orUse":null,"parts":[{"id":"49677712-9c09-437f-8f0b-d146d614f5c0","label":"0","content":"Solve $2^x = 32$","markscheme":"- Recognize $32 = 2^5$ or use logs M1\n- $x = 5$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Identifies the power of 2 or uses logs.","code":"M"},{"id":"p2","marks":1,"description":"Final answer 5.","code":"A"}],"finalAnswers":["5"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"logarithms","label":"Use logarithms","steps":[{"type":"text","order":0,"title":"Take logarithms","content":"Take natural logarithms of both sides. Choosing to use logarithms is a valid method and earns **M1**.","highlights":[{"text":"$$2^x = 32$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\ln\\!\\left(2^x\\right)=\\ln 32$"},{"latex":"$$x\\ln 2=\\ln 32$"}]},"order":1,"title":"Apply the power law","description":"Use the power law to bring the exponent down in front."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x\\ln 2=\\ln 32$"},{"latex":"$$x=\\dfrac{\\ln 32}{\\ln 2}$"},{"latex":"$$x=5$"}]},"order":2,"title":"Divide and evaluate","description":"Divide by ln 2. Evaluating the quotient gives x = 5, earning A1."}]},{"id":"common_base","label":"Write both sides with base 2","steps":[{"type":"text","order":0,"title":"Rewrite the power","content":"Recognise that $$32=2^5.$$ Writing $32$ as a power of $2$ earns **M1**.","highlights":[{"text":"32","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"2^x=32"},{"latex":"2^x=2^5"},{"latex":"x=5"}]},"order":1,"title":"Match the exponents","description":"Rewrite 32 as a power of 2. The bases now match, so equate the exponents. This gives x = 5, earning A1."}]}],"generatedAt":"2026-07-20T16:42:40.505Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":65}},{"id":"0504f7aa-8e4b-4c41-99aa-41eab8797491","revisionId":"fd410dfe-b495-4d4c-94c8-cc3348b85dc6","questionNumber":"4","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","totalMarks":4,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":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 e

1.5 Exponentials and logarithms

Question 2

1.5 Exponentials and logarithms Questions

1.5 Exponentials and logarithms

Question 2

1.5 Exponentials and logarithms Questions