itle":"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","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}},{"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,"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"},"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

Exponentials and Logarithms

Question 4

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 4

Exponentials and Logarithms Questions