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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},{"id":"88185f03-c7d9-4d3b-8a62-0dec9d95df5f","revisionId":"1ce694a2-7f78-4821-8338-eeb94725eb41","questionNumber":"7","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"isCalculation":true,"parts":[{"id":"c79c6280-afda-4e84-8b17-0a74abda9227","label":"0","content":"Solve the equation\n\n$$\\ln(2x + 5) = 1$$","markscheme":"- Remove ln: $2x + 5 = e^1$ M1\n- Rearrange: $2x = e - 5$ M1\n- $x = \\frac{e - 5}{2}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Converts ln to exponential form $e$.","code":"M"},{"id":"p2","marks":1,"description":"Isolates the $x$ term.","code":"M"},{"id":"p3","marks":1,"description":"Final answer $\\frac{e-5}{2}$.","code":"A"}],"finalAnswers":["\\frac{e-5}{2}"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"inverse_exponential","label":"Use the inverse exponential","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=original}{\\ln(2x+5)=1}}$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=exponentiated}{e^{\\ln(2x+5)}=e^1}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=inverted}{2x+5=e^1}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"original"},{"color":"orange","style":"text-color","anchor":"exponentiated"},{"color":"green","style":"text-color","anchor":"inverted"}],"connections":[{"to":"exponentiated","from":"original","color":"orange","label":"exponentiate both sides"},{"to":"inverted","from":"exponentiated","color":"green","label":"inverse functions"}]},"order":1,"title":"Remove the logarithm","description":"The natural logarithm and the exponential function are inverses, so apply the exponential function to both sides. Obtaining 2x + 5 = e^1 earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=before_subtract}{2x+5=e}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=after_subtract}{2x=e-5}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"before_subtract"},{"color":"green","style":"text-color","anchor":"after_subtract"}],"connections":[{"to":"after_subtract","from":"before_subtract","color":"red","label":"subtract 5"}]},"order":2,"title":"Subtract 5","description":"Since e^1 = e, subtract 5 from both sides. Reaching the final line earns the second method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=before_divide}{2x=e-5}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=after_divide}{x=\\dfrac{e-5}{2}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"before_divide"},{"color":"green","style":"text-color","anchor":"after_divide"}],"connections":[{"to":"after_divide","from":"before_divide","color":"green","label":"divide both sides by 2"}]},"order":3,"title":"Divide by two","description":"Divide both sides by 2. The exact solution earns A1; there is no need to replace e with a decimal."}]}],"generatedAt":"2026-07-20T16:27:19.076Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":76},"walkthroughVideo":null},{"id":"ea15c29b-d0c6-4762-aef7-bd0945d0dec2","revisionId":"4a963e32-8652-4688-842c-f6ce007ff0f8","questionNumber":"8","questionType":"LA","specification":"Given that $y = \\log_2 x$, find expressions in terms of $y$ for","difficulty":2,"difficultyBand":"easy","totalMarks":6,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"isCalculation":true,"parts":[{"id":"72ca26f5-371e-4ac7-a9ff-8267ccbd2ac5","label":"a","content":"$$\\log_2 x^2$","markscheme":"- Use power law: $2 \\log_2 x$ M1\n- Substitute $y$: $2y$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Applies power law.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $2y$.","code":"A"}],"finalAnswers":["2y"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"power_law","label":"Use the logarithm power law","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\log_2 \\htmlData{anchor=power}{x^2}$"},{"latex":"$$\\htmlData{anchor=result}{2\\log_2 x}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"result"}],"connections":[{"to":"result","from":"power","color":"sky","label":"power law"}]},"order":1,"title":"Apply the power law","description":"Use the power law: move the exponent inside a logarithm to the front as a multiplier. Applying it correctly here earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$2\\color{@orange}{\\htmlData{anchor=given_log}{\\log_2 x}}$"},{"latex":"$$2\\color{@green}{\\htmlData{anchor=sub_y}{y}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"given_log"},{"color":"green","style":"text-color","anchor":"sub_y"}],"connections":[{"to":"sub_y","from":"given_log","color":"green","label":"substitute"}]},"order":2,"title":"Substitute for the logarithm","description":"The question defines the logarithm as y, so replace the whole logarithm with y. This earns A1."}]}],"generatedAt":"2026-07-20T16:22:26.675Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"3ed3b1bc-70c3-4acd-9c35-e3d686a3590d","label":"b","content":"$$\\log_2 2x$","markscheme":"- Use addition law: $\\log_2 x + \\log_2 2$ M1\n- Simplify: $y + 1$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Applies addition law.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $y+1$.","code":"A"}],"finalAnswers":["y+1"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]},{"id":"af900bab-f5d2-47a8-9918-9684e53df74d","label":"c","content":"$$\\log_8 x$","markscheme":"- Use change of base: $\\frac{\\log_2 x}{\\log_2 8}$ M1\n- Simplify: $\\frac{y}{3}$ A1","marks":2,"order":2,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Applies change of base formula.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $\\frac{y}{3}$.","code":"A"}],"finalAnswers":["y/3"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"change_of_base","label":"Change of base","steps":[{"type":"text","order":0,"title":"Choose base two","content":"Use the change-of-base formula with base $2$, because the given information is $y=\\log_2 x$.","highlights":[{"text":"Given that $y = \\log_2 x$","location":"specification"},{"text":"$$\\log_8 x$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\log_8 x=\\dfrac{\\color{@sky}{\\htmlData{anchor=given_log}{\\log_2 x}}}{\\log_2 8}$"},{"latex":"$$\\log_2 8=\\color{@amber}{3}$"},{"latex":"$$\\log_8 x=\\dfrac{\\color{@sky}{\\htmlData{anchor=result_y}{y}}}{\\color{@amber}{3}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"given_log"},{"color":"sky","style":"text-color","anchor":"result_y"}],"connections":[{"to":"result_y","from":"given_log","color":"sky","label":"given as y"}]},"order":1,"title":"Change the base","description":"The change-of-base quotient earns M1. Using the fact that 8 is 2 cubed, so the denominator is 3, and simplifying to y/3 earns A1."}]}],"generatedAt":"2026-07-20T16:42:18.892Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":46},"walkthroughVideo":null}],"subjectCode":"a_level_maths","summaries":[{"id":"5e711279-248d-4041-8b5b-944dc2a079e2","difficultyBand":null,"calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"71ec8bdb-d5b0-433a-9316-f3aafebdd276","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"66787241-20fe-4677-a28d-e2080738231a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"ac0c508c-786a-4c5a-b076-8d1fca860f98","difficultyBand":null,"calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"9637e647-82d8-412c-ab92-13583c6f5d65","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e5a61db3-1c05-4d3f-982e-8cb54f173814","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"0504f7aa-8e4b-4c41-99aa-41eab8797491","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"100baeb7-11ee-44ce-8593-abbac07096d8","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c1524aa2-f8bc-4f32-89b3-4ab29cc37f0f","difficultyBand":null,"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":"e4f7a0f1-0bc3-4b5c-8cd1-e6acc6abbaa6","difficultyBand":null,"calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"feae588b-ccc6-49d9-910b-e2c20ede5492","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"297a6c62-3939-4bf2-b695-5abc3e67f5c7","difficultyBand":null,"calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"88185f03-c7d9-4d3b-8a62-0dec9d95df5f","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a58044a8-a4d6-48ff-96df-48be4f3ec4d1","difficultyBand":null,"calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"ea15c29b-d0c6-4762-aef7-bd0945d0dec2","difficultyBand":"easy","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":"c6a2be8b-037e-4010-ac66-2e0358c65a4f","difficultyBand":null,"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":"54a4c1e3-f02f-480a-9b20-c0b1f667e828","difficultyBand":null,"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":"fd00b03b-1e6d-4862-906a-adce6e680d80","difficultyBand":null,"calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"3fb0edd5-94e8-4403-8a78-34fe1055baf5","difficultyBand":null,"calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"9d18c848-dc6f-45d9-b2ae-030c82d05813","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"4f882717-8c90-4f68-af94-914e8330f88f","difficultyBand":null,"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":"29663621-97bc-41a3-a277-f410fef439c6","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"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-abce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and Logarithms","topicId":"c1163ef0-9f94-ec55-d543-6ee688ac4778","topicSidebar":"$L3c","topicSlug":"as-pure-mathematics-14-exponentials-and-logarithms","topicTitle":"Exponentials and Logarithms"}]}],"$L3d","$L3e"]}]

Exponentials and Logarithms

Question 5

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 5

Exponentials and Logarithms Questions