:null,"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"},"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}},{"id":"f41109a7-42e0-417b-8d30-f489b624e4a7","revisionId":"5cc9c054-b1e5-4167-97d1-a997f0b7d107","questionNumber":"5","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"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"},"explanation":{"methods":[{"id":"logarithm_laws","label":"Apply the logarithm laws","steps":[{"type":"hyper_latex","b

1.5 Exponentials and logarithms

Question 3

1.5 Exponentials and logarithms Questions

1.5 Exponentials and logarithms

Question 3

1.5 Exponentials and logarithms Questions