"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":"53f29878-46de-467c-aba0-8f24236f072a","revisionId":"da1193c1-2593-48c4-8cd7-11fd52727b55","questionNumber":null,"questionType":"LA","specification":"The total cost, $C$, of manufacturing $q$ units of a specialized product is modeled by the equation $C = aq^n$, where $a$ and $n$ are constants.","difficulty":2,"difficultyBand":"easy","totalMarks":3,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"13bbeab7-4a8c-4815-b259-f0e2607f811c","label":"a","content":"Show that this relationship can be written in the form $\\log C = \\log a + n \\log q$.","markscheme":"- $\\log C = \\log aq^n$ M1\n- $\\log C = \\log a + \\log q^n$ M1\n- $\\log C = \\log a + n \\log q$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Takes the logarithm of both sides of the equation.","code":"M","acceptable":["$$\\log C = \\log(aq^n)$"]},{"id":"p2","marks":1,"description":"Applies the log addition law.","code":"M","acceptable":["$$\\log C = \\log a + \\log q^n$"]},{"id":"p3","marks":1,"description":"Applies the log power law to reach the final form.","code":"A"}],"finalAnswers":["$$\\log C = \\log a + n \\log q$"],"generatedAt":"2026-05-06T14:05:19.695Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"direct_derivation","label":"Direct Logarithmic Expansion","steps":[{"type":"text","order":0,"title":"Take logarithms of both sides","content":"We start with the given equation:\\n\\n$$C = aq^n$$\\n\\nTo begin our expansion, we take the logarithm of both sides of the equation:\\n\\n$$\\log C = \\log(aq^n)$$\\n\\nThis first step earns the first method mark $\\text{[M1]}$."},{"type":"text","order":1,"title":"Apply the product rule","content":"Next, we use the product law of logarithms, which states that $\\log(xy) = \\log x + \\log y$.\\n\\nApplying this to the right-hand side of our equation, we can split the term $\\log(aq^n)$:\\n\\n$$\\log C = \\log a + \\log(q^n)$$\\n\\nShowing this step earns the second method mark $\\text{[M1]}$."},{"type":"text","order":2,"title":"Apply the power rule","

1.6 Exponentials and Logarithms

Question 2

1.6 Exponentials and Logarithms Questions

1.6 Exponentials and Logarithms

Question 2

1.6 Exponentials and Logarithms Questions