37:I[651451,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"UserProgressProvider"] 38:I[196793,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"Gingerbread"] 3a:I[716297,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"TopicRecentlyViewed"] 3b:I[647935,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"RecordResourceResume"] 3c:I[426249,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"TopicQuestionBank"] 25:["$","$L37",null,{"bundle":{"topicIds":["ab4b4b13-1cde-4ca0-b679-5750ea506ee7","0b4dedaf-093b-4513-b270-5ef5f1799402","e823de84-4e2c-4e0d-9ef8-76030c6e16d4","64e202bf-d7c8-445a-9d80-2875f8dda786","3d3ef618-222c-4891-a274-22c152dfb1a0","77e2af3b-4aaa-4111-87a8-963ca79d763e","4baaad32-cd8e-4aca-860e-9eea09e614bd","af007230-3b7d-45b5-b9f9-ff7520278316","a2a3fac4-6de1-4c0f-b1f1-b3587fb20014","d9ac091b-b628-4ed1-8008-a5793781f003","c48492d8-bdec-4713-87cf-06c3e1af41c8","eb9d3953-2fda-4e93-8787-8d2a634586b6"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L38",null,{"feature":"question-bank"}],"$L39",["$","$L3a",null,{"subjectCode":"a_level_maths__ocr_mei","subjectName":"Maths","topicSlug":"exponentials-and-logarithms","topicTitle":"1.8 Exponentials and Logarithms","sectionTitle":null,"activity":"worksheet","resourceTitle":"1.8 Exponentials and Logarithms Questions","resourceHref":"/a-level/maths/ocr/mei/exponentials-and-logarithms/questions/q1","saveTrail":false}],["$","$L3b",null,{"href":"/a-level/maths/ocr/mei/exponentials-and-logarithms/questions/q1","label":"1.8 Exponentials and Logarithms","subjectCode":"a_level_maths__ocr_mei","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 15 – 1.8 Exponentials and Logarithms - Questions\",\"description\":\"Practise 1.8 Exponentials and Logarithms A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"1.8 Exponentials and Logarithms\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/ocr/mei/exponentials-and-logarithms/questions/q15\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L3c",null,{"backHref":"/a-level/maths/ocr/mei/questions","basePath":"/a-level/maths/ocr/mei/exponentials-and-logarithms/questions","formulaUrl":null,"initialIndex":14,"initialQuestionWindow":[{"id":"53f29878-46de-467c-aba0-8f24236f072a","revisionId":"da1193c1-2593-48c4-8cd7-11fd52727b55","questionNumber":null,"questionType":"LA","specification":"The total cost, $C$, of manufacturing $q$ units of a specialized product is modeled by the equation $C = aq^n$, where $a$ and $n$ are constants.","difficulty":2,"difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"13bbeab7-4a8c-4815-b259-f0e2607f811c","label":"a","content":"Show that this relationship can be written in the form $\\log C = \\log a + n \\log q$.","markscheme":"- $\\log C = \\log aq^n$ M1\n- $\\log C = \\log a + \\log q^n$ M1\n- $\\log C = \\log a + n \\log q$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Takes the logarithm of both sides of the equation.","code":"M","acceptable":["$$\\log C = \\log(aq^n)$"]},{"id":"p2","marks":1,"description":"Applies the log addition law.","code":"M","acceptable":["$$\\log C = \\log a + \\log q^n$"]},{"id":"p3","marks":1,"description":"Applies the log power law to reach the final form.","code":"A"}],"finalAnswers":["$$\\log C = \\log a + n \\log q$"],"generatedAt":"2026-05-06T14:05:19.695Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_derivation","label":"Direct Logarithmic Expansion","steps":[{"type":"text","order":0,"title":"Take logarithms of both sides","content":"We start with the given equation:\\n\\n$$C = aq^n$$\\n\\nTo begin our expansion, we take the logarithm of both sides of the equation:\\n\\n$$\\log C = \\log(aq^n)$$\\n\\nThis first step earns the first method mark $\\text{[M1]}$."},{"type":"text","order":1,"title":"Apply the product rule","content":"Next, we use the product law of logarithms, which states that $\\log(xy) = \\log x + \\log y$.\\n\\nApplying this to the right-hand side of our equation, we can split the term $\\log(aq^n)$:\\n\\n$$\\log C = \\log a + \\log(q^n)$$\\n\\nShowing this step earns the second method mark $\\text{[M1]}$."},{"type":"text","order":2,"title":"Apply the power rule","content":"Finally, we apply the power law of logarithms, which states that $\\log(x^k) = k \\log x$.\\n\\nApplying this to the term $\\log(q^n)$ allows us to bring the exponent $n$ to the front:\\n\\n$$\\log C = \\log a + n \\log q$$\\n\\nThis completes the proof and earns the final accuracy mark $\\text{[A1]}$."}]},{"id":"reverse_derivation","label":"Reverse Combination","steps":[{"type":"text","order":0,"title":"Start with target expression","content":"To prove the statement using the **Reverse Combination** approach, we begin with the right-hand side (RHS) of the given target equation:\\n\\n$$\\text{RHS} = \\log a + n \\log q$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\log a + \\htmlData{anchor=orig-pow}{\\color{@sky}{n \\log q}}$"},{"latex":"$$\\log a + \\htmlData{anchor=new-pow}{\\color{@sky}{\\log q^n}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"orig-pow"},{"color":"sky","style":"text-color","anchor":"new-pow"}],"connections":[{"to":"new-pow","from":"orig-pow","color":"sky","label":"power law","style":"solid"}]},"order":1,"title":"Apply the power law","description":"First, we apply the power law of logarithms, $$k \\log x = \\log(x^k)$$, to rewrite the second term of our expression."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=term1}{\\color{@purple}{\\log a}} + \\htmlData{anchor=term2}{\\color{@purple}{\\log q^n}}$"},{"latex":"$$\\htmlData{anchor=combined}{\\color{@purple}{\\log(a q^n)}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"term1"},{"color":"purple","style":"text-color","anchor":"term2"},{"color":"purple","style":"text-color","anchor":"combined"}],"connections":[{"to":"combined","from":"term1","color":"purple","label":"product law","style":"solid"}]},"order":2,"title":"Combine terms with product law","description":"Next, we use the addition law (product law) of logarithms, $$\\log x + \\log y = \\log(xy)$$, to combine the two logarithmic terms into a single logarithm."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$C = \\htmlData{anchor=c-val}{\\color{@teal}{aq^n}}$"},{"latex":"$$\\log(\\htmlData{anchor=c-sub}{\\color{@teal}{aq^n}})$"},{"latex":"$$\\log \\htmlData{anchor=final-c}{\\color{@teal}{C}}$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"c-val"},{"color":"teal","style":"text-color","anchor":"c-sub"},{"color":"teal","style":"text-color","anchor":"final-c"}],"connections":[{"to":"c-sub","from":"c-val","color":"teal","label":"substitute","style":"solid"}]},"order":3,"title":"Substitute the original model","description":"Since the question states that the manufacturing model is given by $$C = aq^n$$, we can replace the expression inside the logarithm with $$C$$."},{"type":"text","order":4,"title":"Conclude the proof","content":"By simplifying the right-hand side, we have shown that:\\n\\n$$\\log a + n \\log q = \\log C$$\\n\\nRearranging this gives our desired form:\\n\\n$$\\log C = \\log a + n \\log q$$\\n\\n### Markscheme Breakdown\\n- **M1**: Establishing the connection to the single logarithm form $\\log C = \\log(aq^n)$.\\n- **M1**: Using the multiplication law to split the logarithmic terms: $\\log C = \\log a + \\log(q^n)$.\\n- **A1**: Correctly applying the power law to obtain the final relationship: $\\log C = \\log a + n \\log q$."}]}],"generatedAt":"2026-06-09T01:59:32.456Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"13bbeab7-4a8c-4815-b259-f0e2607f811c:direct_derivation","label":"Direct Logarithmic Expansion","steps":[{"type":"text","order":0,"title":"Take logarithms of both sides","content":"We start with the given equation:\\n\\n$$C = aq^n$$\\n\\nTo begin our expansion, we take the logarithm of both sides of the equation:\\n\\n$$\\log C = \\log(aq^n)$$\\n\\nThis first step earns the first method mark $\\text{[M1]}$."},{"type":"text","order":1,"title":"Apply the product rule","content":"Next, we use the product law of logarithms, which states that $\\log(xy) = \\log x + \\log y$.\\n\\nApplying this to the right-hand side of our equation, we can split the term $\\log(aq^n)$:\\n\\n$$\\log C = \\log a + \\log(q^n)$$\\n\\nShowing this step earns the second method mark $\\text{[M1]}$."},{"type":"text","order":2,"title":"Apply the power rule","content":"Finally, we apply the power law of logarithms, which states that $\\log(x^k) = k \\log x$.\\n\\nApplying this to the term $\\log(q^n)$ allows us to bring the exponent $n$ to the front:\\n\\n$$\\log C = \\log a + n \\log q$$\\n\\nThis completes the proof and earns the final accuracy mark $\\text{[A1]}$."}]},{"id":"13bbeab7-4a8c-4815-b259-f0e2607f811c:reverse_derivation","label":"Reverse Combination","steps":[{"type":"text","order":0,"title":"Start with target expression","content":"To prove the statement using the **Reverse Combination** approach, we begin with the right-hand side (RHS) of the given target equation:\\n\\n$$\\text{RHS} = \\log a + n \\log q$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\log a + \\htmlData{anchor=orig-pow}{\\color{@sky}{n \\log q}}$"},{"latex":"$$\\log a + \\htmlData{anchor=new-pow}{\\color{@sky}{\\log q^n}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"orig-pow"},{"color":"sky","style":"text-color","anchor":"new-pow"}],"connections":[{"to":"new-pow","from":"orig-pow","color":"sky","label":"power law","style":"solid"}]},"order":1,"title":"Apply the power law","description":"First, we apply the power law of logarithms, $$k \\log x = \\log(x^k)$$, to rewrite the second term of our expression."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=term1}{\\color{@purple}{\\log a}} + \\htmlData{anchor=term2}{\\color{@purple}{\\log q^n}}$"},{"latex":"$$\\htmlData{anchor=combined}{\\color{@purple}{\\log(a q^n)}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"term1"},{"color":"purple","style":"text-color","anchor":"term2"},{"color":"purple","style":"text-color","anchor":"combined"}],"connections":[{"to":"combined","from":"term1","color":"purple","label":"product law","style":"solid"}]},"order":2,"title":"Combine terms with product law","description":"Next, we use the addition law (product law) of logarithms, $$\\log x + \\log y = \\log(xy)$$, to combine the two logarithmic terms into a single logarithm."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$C = \\htmlData{anchor=c-val}{\\color{@teal}{aq^n}}$"},{"latex":"$$\\log(\\htmlData{anchor=c-sub}{\\color{@teal}{aq^n}})$"},{"latex":"$$\\log \\htmlData{anchor=final-c}{\\color{@teal}{C}}$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"c-val"},{"color":"teal","style":"text-color","anchor":"c-sub"},{"color":"teal","style":"text-color","anchor":"final-c"}],"connections":[{"to":"c-sub","from":"c-val","color":"teal","label":"substitute","style":"solid"}]},"order":3,"title":"Substitute the original model","description":"Since the question states that the manufacturing model is given by $$C = aq^n$$, we can replace the expression inside the logarithm with $$C$$."},{"type":"text","order":4,"title":"Conclude the proof","content":"By simplifying the right-hand side, we have shown that:\\n\\n$$\\log a + n \\log q = \\log C$$\\n\\nRearranging this gives our desired form:\\n\\n$$\\log C = \\log a + n \\log q$$\\n\\n### Markscheme Breakdown\\n- **M1**: Establishing the connection to the single logarithm form $\\log C = \\log(aq^n)$.\\n- **M1**: Using the multiplication law to split the logarithmic terms: $\\log C = \\log a + \\log(q^n)$.\\n- **A1**: Correctly applying the power law to obtain the final relationship: $\\log C = \\log a + n \\log q$."}]}],"generatedAt":"2026-06-09T01:59:53.025Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"0504f7aa-8e4b-4c41-99aa-41eab8797491","revisionId":"fd410dfe-b495-4d4c-94c8-cc3348b85dc6","questionNumber":"4","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"isCalculation":true,"parts":[{"id":"21d71d58-c380-47a2-bbb7-eb98c61b8ee5","label":"0","content":"Solve the equation\n\n$$\\log_3(x) + \\log_3(4) = 2$$","markscheme":"- Use addition law: $\\log_3 4x = 2$ M1\n- Remove logs: $3^2 = 4x$ M1\n- $9 = 4x$ M1\n- $x = \\frac{9}{4}$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Combines logs using addition law.","code":"M"},{"id":"p2","marks":1,"description":"Converts to exponential form $3^2 = 4x$.","code":"M"},{"id":"p3","marks":1,"description":"Simplifies $3^2$ to 9.","code":"M"},{"id":"p4","marks":1,"description":"Final answer $2.25$ or $\\frac{9}{4}$.","code":"A"}],"finalAnswers":["2.25","9/4"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"logarithm_law_fixed","label":"Use logarithm laws","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=logsum}{\\color{@sky}{\\log_3(x)+\\log_3(4)}}=2$"},{"latex":"$$\\log_3(\\htmlData{anchor=product}{\\color{@sky}{4x}})=2\\quad\\text{(M1)}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"logsum"},{"color":"sky","style":"text-color","anchor":"product"}]},"order":0,"title":"Combine the logarithms","description":"Combine the logarithms using the product law. This earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=logform}{\\color{@orange}{\\log_3(4x)=2}}$"},{"latex":"$$\\htmlData{anchor=argument}{\\color{@orange}{3^2=4x}}\\quad\\text{(M1)}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"logform"},{"color":"orange","style":"text-color-with-background","anchor":"argument"}]},"order":1,"title":"Use exponential form","description":"Rewrite log₃(4x) = 2 in exponential form. This earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=power}{\\color{@green}{3^2=4x}}$"},{"latex":"$$\\htmlData{anchor=evaluated}{\\color{@green}{9=4x}}\\quad\\text{(M1)}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"power"},{"color":"green","style":"text-color-with-background","anchor":"evaluated"}]},"order":2,"title":"Evaluate the power","description":"Evaluate 3² to obtain 9 = 4x. This earns the third M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=coefficient}{\\color{@sky}{9=4x}}$"},{"latex":"$$\\htmlData{anchor=answer}{\\color{@green}{x=\\frac{9}{4}}}\\quad\\text{(A1)}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"coefficient"},{"color":"green","style":"text-color-with-background","anchor":"answer"}]},"order":3,"title":"Isolate the variable","description":"Divide by 4 to make x the subject. This earns the final A1."}]}],"generatedAt":"2026-07-20T16:29:18.126Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":69},"walkthroughVideo":null},{"id":"f41109a7-42e0-417b-8d30-f489b624e4a7","revisionId":"5cc9c054-b1e5-4167-97d1-a997f0b7d107","questionNumber":"5","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","tier":null,"totalMarks":2,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"isCalculation":true,"parts":[{"id":"badfcafa-8d2b-4f52-8a39-b931b12bc93c","label":"0","content":"Express as a single logarithm to base $a$\n\n$$2\\log_a(x + 1) - \\log_a(4)$$","markscheme":"- Use power law: $\\log_a(x+1)^2 - \\log_a 4$ M1\n- Use subtraction law: $\\log_a\\left(\\frac{(x+1)^2}{4}\\right)$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":2,"description":"Applies power law to the first term.","code":"M"},{"id":"p2","marks":2,"description":"Applies subtraction law to combine into a single log.","code":"A"}],"finalAnswers":["\\log_a(\\frac{(x+1)^2}{4})"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"logarithm_laws","label":"Apply the logarithm laws","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=coefficient}{2}}\\log_a(x+1)-\\log_a(4)$"},{"latex":"$$\\log_a\\!\\left((x+1)^{\\color{@sky}{\\htmlData{anchor=exponent}{2}}}\\right)-\\log_a(4)$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"coefficient"},{"color":"sky","style":"text-color","anchor":"exponent"}]},"order":0,"title":"Move the coefficient","description":"Apply the power law to move the coefficient into the logarithm as an exponent. This earns the method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\log_a\\left((x+1)^2\\right)-\\color{@green}{\\htmlData{anchor=subtracted_log}{\\log_a(4)}}$"},{"latex":"$$\\log_a\\left(\\dfrac{(x+1)^2}{\\color{@green}{\\htmlData{anchor=denominator}{4}}}\\right)$"}],"highlights":[{"color":"green","style":"text-color","anchor":"subtracted_log"},{"color":"green","style":"text-color","anchor":"denominator"}]},"order":1,"title":"Combine by division","description":"A subtraction of logarithms becomes the logarithm of a quotient. This gives the required single logarithm and earns A1."}]}],"generatedAt":"2026-07-20T16:27:04.614Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":79},"walkthroughVideo":null}],"subjectCode":"a_level_maths__ocr_mei","summaries":[{"id":"71ec8bdb-d5b0-433a-9316-f3aafebdd276","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"a5ebe963-bbe5-43b3-b2cd-f01bd32e40d8","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"b4ac2bbe-3a73-41eb-9bb2-cd5c86bea13c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d4762c54-72db-4c30-a8a5-6ac7eadbc5b6","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"d6c2938e-d074-4563-a7f7-483d487e4f7d","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"66787241-20fe-4677-a28d-e2080738231a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"37af4f21-ab7f-40db-93f6-5149b192c675","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"53f29878-46de-467c-aba0-8f24236f072a","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6e05362c-a0d3-4fe6-aed9-1a4786aa7dc1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9637e647-82d8-412c-ab92-13583c6f5d65","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"4b4b0637-07ef-4c08-8d71-a068ce5be4bd","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"4d6c02bc-272a-4b6f-9f68-e4786589e216","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"5760e0f1-89f1-47aa-920c-5c7f6a19d16c","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"8d490b61-23c6-483c-b67c-5ced33bb3d08","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"0504f7aa-8e4b-4c41-99aa-41eab8797491","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f41109a7-42e0-417b-8d30-f489b624e4a7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"28d128cf-7f7e-49cd-a7dd-4f385cbc97fa","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"feae588b-ccc6-49d9-910b-e2c20ede5492","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"88185f03-c7d9-4d3b-8a62-0dec9d95df5f","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b899b379-6149-43e6-ab2c-07612ef34732","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"13a3c314-7e81-4552-ac4c-cabbed257c80","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ea15c29b-d0c6-4762-aef7-bd0945d0dec2","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0052236a-8757-4305-86cd-686424ae9eb8","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9434c3ed-c63c-4953-811c-6170426c7815","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"516d0a9c-9e77-486d-97a5-47044f123d7b","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"64ea9803-5740-4af1-9e60-de02022423cb","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9d18c848-dc6f-45d9-b2ae-030c82d05813","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"c213fe79-05e5-4985-9497-74ee6ceed769","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"afdf1b4b-d6cb-43ac-9c97-67360967909c","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"793c92a1-46cb-4a81-a3e6-121c2bbdfb57","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"efb9f39e-5783-4f3d-b0ff-8cb66a8a15f6","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"a0b6a559-f180-4915-9c95-f8c861f5c721","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"1bfcb2ae-52cb-4aca-9e4c-af12063044cd","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"ec9dd2f1-a8eb-4350-9d79-abce80fcda41","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"22679cf4-649a-4d98-be19-7daf2bdf3cd0","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"261ef45d-7b83-4cf5-884b-3a2e11deeff3","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"b520f7da-b834-40b6-821e-c3c7ecc5c314","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"119996f2-9c31-4cd5-a233-1319e0864977","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"0e1ac3b1-a875-4c7e-a165-07f14d2520de","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"c97e3f1c-8489-4584-bcc5-3801e0db88fb","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"d2bdfd8c-98fc-4b0b-ba83-5113ad258ed0","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"7c42e3fd-0056-45cc-b061-d475d97fa7a1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9d34b926-3415-45c1-86e1-505a718cb27c","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"cf13a964-5be5-44fe-bca6-ff41b1c57712","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"2bda2fce-f999-40c8-a235-35ce417301b7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"0056c4bc-8954-411c-b6d4-5e82a963d6f5","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b1f3d1e7-530c-485e-a178-32ee16a6d4bf","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"984fb8ce-5927-4ec6-9209-384c495bf0aa","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"79c13365-8bcd-445a-87de-7fbc150d7627","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b11d3008-1aa2-4c18-88c5-b5d3945c0713","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"9d9d38e1-ea87-493e-9654-ae3f321fb6af","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c30411eb-1d7a-4553-b6a4-0dbb39e6f514","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"0acfafca-9381-4a74-afe7-7e60dbcf5223","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"0d755b15-99c3-4016-a0da-d456dace4435","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"0d780a9c-6092-425b-8de8-2c57bc7d0052","difficultyBand":"hard","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1121fd8f-3a29-4838-883a-b5b9938633a7","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"132b8b62-bcee-4b73-a053-8bb983eeffa1","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"1a483881-a10f-45a4-98a1-ce4dd4bcd885","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"1f0481c4-f42c-4d9b-9705-019fa4687d6e","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"284bd084-87d9-4be0-bffa-bff2f355cf1e","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"2b481c4e-995e-4dab-b0e9-3f05d0c18ef0","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"32fdf593-88a0-4dbb-aa73-ad860b71b919","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"3523c14c-f2b6-45be-9e28-c1c6ccd76d12","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"35eb1b34-5b3b-4128-af99-4fad5a651a03","difficultyBand":"hard","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"380f0bd4-4fa2-407d-bc2c-583573c54d12","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"5263cd47-c4ca-4a3c-914f-27c96aa220f0","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"52a6ef15-d581-44e4-83f1-21f3581f0dc1","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"52e25ef1-0098-4297-a077-6eba69984c8e","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"56b53b85-c9cb-490f-ad3e-0b48e8fe203b","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"5b84449f-4dc5-4c79-b8b4-9f1ac7b1e555","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"5ca71b91-2d4f-4227-89e9-e044ca6d3e50","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"66b65989-8876-416f-98ef-a072de45e87c","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"73e44ee1-65ef-4d0d-a4b3-323428c62a77","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"743c75f8-c7f4-45e0-990d-487ee4004420","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"777b6e71-0785-4eb9-8d96-fac0b374d21a","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"77877a2f-972d-4373-b278-05e29ec6cdb9","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"8579a062-ff0e-48ab-b137-eb10b474fd98","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"875eb0c9-b673-4641-9ac8-09ca6fab6e8b","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":false},{"id":"8a829253-f256-4624-9f94-b3f17e42ad57","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"8ad866db-5e80-4066-b7d3-cd01be06e643","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"8dc70a58-256a-4b59-8030-870b7d94941c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"949298be-384c-4dae-a3ed-3c4cbe195a14","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"95f5cf7e-ecc4-4939-9ff5-389864b282f3","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"963cde07-e595-410a-8297-f7dae2370e09","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9ba1d30a-bc4a-468c-9b76-b270c569f4a9","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"9cb41ac2-1773-4c05-8a10-980c6f714930","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"9cede472-3641-4574-b054-1559b96e50fc","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"9fe0c6f2-be69-4610-a247-12db900fd3d9","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"a00a3ab0-9f29-4b00-8f33-da1f0dfd3ca5","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"ad7be240-3211-45ab-81d5-d8ea26d20f6f","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"b1fc3b42-e7ca-4510-9e5f-72310d85ffe9","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"bb050852-0193-4c40-aa78-1ae7e7672d60","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"bd601092-2681-4e64-bbb0-0c9a787e5c62","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"bee7e273-893c-4468-9b02-399fec63f8ba","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"c49c836a-c26e-4d9d-a179-f703bc1028cb","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"ce74b90a-937a-411e-a33a-317aef1a9ef1","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"d95e6b03-a250-4760-997f-2244154bd743","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"dbec8508-1874-4205-b3f2-a31fdcd0f964","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":true},{"id":"dec2be4b-105e-49de-b60a-db1c4e9664b1","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e3469d06-b5f3-47da-af25-4ecb218ce61d","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"f5be14d6-f018-4715-a183-64043a01cd27","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"f6385dd7-cb6d-44c0-bbaf-3f3540ffafed","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"f9fb84bb-518c-411b-b131-c6f954bd77ce","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fdc0d761-5a31-4d9e-9e60-de5eda974e41","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true}],"title":"1.8 Exponentials and Logarithms","topicId":"ab4b4b13-1cde-4ca0-b679-5750ea506ee7","topicSidebar":"$L3d","topicSlug":"exponentials-and-logarithms","topicTitle":"1.8 Exponentials and Logarithms"}]}],"$L3e","$L3f"]}]

1.8 Exponentials and Logarithms

Question 4

1.8 Exponentials and Logarithms Questions

1.8 Exponentials and Logarithms

Question 4

1.8 Exponentials and Logarithms Questions