ms, then divide both sides by negative 0.15 before evaluating. Obtaining t = 8.50 minutes earns the accuracy mark (A1)."}]}],"generatedAt":"2026-07-21T04:10:44.275Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"1bfcb2ae-52cb-4aca-9e4c-af12063044cd","revisionId":"04555a9f-6527-4cd3-9139-6dd8a855a442","questionNumber":"19","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"parts":[{"id":"ddde3cb7-f0dc-42da-887e-8bf98b322ab0","label":"0","content":"Show that\n\n$$\\log_2\\left(\\frac{x}{8}\\right) + 2\\log_2 3 \\equiv \\log_2(9x) - 3$$","markscheme":"- LHS: $\\log_2 x - \\log_2 8 + \\log_2 3^2$ M1\n- $\\log_2 x - 3 + \\log_2 9$ M1\n- $\\log_2(9x) - 3$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Applies subtraction and power laws to the LHS.","code":"M"},{"id":"p2","marks":1,"description":"Evaluates $\\log_2 8 = 3$.","code":"M"},{"id":"p3","marks":1,"description":"Combines $\\log_2 x$ and $\\log_2 9$ to reach the RHS.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"log_laws","label":"Use logarithm laws","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"\\log_2\\left(\\frac{x}{8}\\right)+2\\log_2 3=\\log_2 x-\\log_2 8+\\htmlData{anchor=power_copy}{2\\log_2 3}"},{"latex":"\\log_2 x-\\log_2 8+2\\log_2 3=\\log_2 x-\\log_2 8+\\htmlData{anchor=power_result}{\\log_2(3^2)}"}],"connections":[{"to":"power_result","from":"power_copy","color":"sky","label":"power law"}]},"order":1,"title":"Apply both laws","description":"The quotient becomes subtraction, and the coefficient becomes an exponent. This correct expansion earns the first method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\log_2 x-\\htmlData{anchor=log8}{\\log_2 8}+\\log_2(\\htmlData{anchor=square}{3^2})"},{"latex":"\\htmlData{anchor=log8_eval}{\\log_2 8=3}"},{"latex":"\\htmlData{anchor=square_eval}{3^2=9}"},{"latex":"\\log_2 x-3+\\log_2 9"}],"connections":[{"to":"log8_eval","from":"log8","color":"sky","label":"evaluate"},{"to":"square_eval","from":"square","color":"orange","label":"square 3"}]},"order":2,"title":"Evaluate the powers","description":"Use 8 = 2 cubed, so log base 2 of 8 is 3, and use 3 squared = 9. These evaluations give the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=whole_before}{\\htmlData{anchor=sum_logs}{\\log_2 x+\\log_2 9}-3}"},{"latex":"\\htmlData{anchor=whole_after}{\\htmlData{anchor=product_result}{\\log_2(9x)}-3}"}],"connections":[{"to":"whole_after","from":"whole_before","color":"green","label":"product law"}]},"order":3,"title":"Combine the logarithms","description":"Reorder the terms so the two logarithms are together, then apply the product law. This gives the required form and earns the final accuracy mark (A1)."}]}],"generatedAt":"2026-07-20T16:45:34.777Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":81}},{"id":"ec9dd2f1-a8eb-4350-9d79-abce80fcda41","revisionId":"079ead9f-20c4-46ac-9f1e-dd2bb2a0d5c0","questionNumber":"20","questionType":"LA","specification":"","difficulty":3,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"parts":[{"id":"8885e823-4a7d-45ff-b794-0e4125ce396f","label":"a","content":"Using $y = 2^x$ as a substitution, show that $4^x - 2^{x+2} - 5 = 0$ can be written as $y^2 - 4y - 5 = 0$.","markscheme":"- $4^x = (2^2)^x = (2^x)^2 = y^2$ B1\n- $2^{x+2} = 2^x \\cdot 2^2 = 4y$ B1\n- Result follows: $y^2 - 4y - 5 = 0$","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Shows $4^x = y^2$.","code":"B

Exponentials and Logarithms

Question 12

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 12

Exponentials and Logarithms Questions