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":85}},{"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"},{"id":"p2","marks":1,"description":"Shows $2^{x+2} = 4y$.","code":"B"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"index_law_substitution","label":"Index laws and substitution","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=first_power}{4^x}=(2^2)^x$"},{"latex":"$$(2^2)^x=(2^x)^2$"},{"latex":"$$(2^x)^2=\\htmlData{anchor=first_result}{y^2}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"first_result"}]},"order":1,"title":"Rewrite the first term","description":"Use the substitution y = 2ˣ, then apply the power-of-a-power index law. This complete chain earns the first B1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$2^{x+2}=\\htmlData{anchor=expanded_term}{2^x\\cdot2^2}$"},{"latex":"$$2^{x+2}=\\htmlData{anchor=second_result}{4y}$"}],"highlights":[{"color":"orange","style":"text-color-with-background","anchor":"second_result"}],"connections":[{"to":"second_result","from":"expanded_term","color":"orange","style":"dashed"}]},"order":2,"title":"Rewrite the second term","description":"Use the index law for adding indices, then use y = 2ˣ. Obtaining 4y earns the second B1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=original_first}{\\color{@sky}{4^x}}-\\htmlData{anchor=original_second}{\\color{@orange}{2^{x+2}}}-5=0$"},{"latex":"$$\\htmlData{anchor=result_first}{\\color{@sky}{y^2}}-\\htmlData{anchor=result_second}{\\color{@orange}{4y}}-5=0$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"original_first"},{"color":"sky","style":"text-color","anchor":"result_first"},{"color":"orange","style":"text-color","anchor":"original_second"},{"color":"orange","style":"text-color","anchor":"result_second"}]},"order":3,"title":"Substitute both results","description":"Replace the two exponential terms with y² and 4y. This gives the required quadratic equation."}]}],"generatedAt":"2026-07-20T16:52:28.020Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"60c07025-410f-4512-9634-2cd9df814da5","label":"b","content":"Hence, show that the equation $4^x - 2^{x+2} - 5 = 0$ has $x = \\log_2 5$ as its only solution.","markscheme":"- Factorise: $(y-5)(y+1) = 0$ M1\n- $y=5$ or $y=-1$ A1\n- $2^x = 5 \\Rightarrow x = \\log_2 5$ M1\n- $2^x = -1$ has no solution as $2^x > 0$ A1","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Solves the quadratic equation for $y$.","code":"M"},{"id":"p2","marks":1,"description":"Identifies $y=5$ and $y=-1$.","code":"A"},{"id":"p3","marks":1,"description":"Converts $y=5$ back to $x = \\log_2 5$.","code":"M"},{"id":"p4","marks":1,"description":"Explains why $y=-1$ yields no solution.","code":"A"}],"finalAnswers":["\\log_2 5"],"generatedAt":"2

Exponentials and Logarithms

Question 10

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 10

Exponentials and Logarithms Questions