torUse":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":"$$\\color{@green}{\\htmlData{anchor=inverted}{2x+5=e^1}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"original"},{"color":"orange","style":"text-color","anchor":"exponentiated"},{"color":"green","style":"text-color","anchor":"inverted"}],"connections":[{"to":"exponentiated","from":"original","color":"orange","label":"exponentiate both sides"},{"to":"inverted","from":"exponentiated","color":"green","label":"inverse functions"}]},"order":1,"title":"Remove the logarithm","description":"The natural logarithm and the exponential function are inverses, so apply the exponential function to both sides. Obtaining 2x + 5 = e^1 earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=before_subtract}{2x+5=e}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=after_subtract}{2x=e-5}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"before_subtract"},{"color":"green","style":"text-color","anchor":"after_subtract"}],"connections":[{"to":"after_subtract","from":"before_subtract","color":"red","label":"subtract 5"}]},"order":2,"title":"Subtract 5","description":"Since e^1 = e, subtract 5 from both sides. Reaching the final line earns the second method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=before_divide}{2x=e-5}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=after_divide}{x=\\dfrac{e-5}{2}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"before_divide"},{"color":"green","style":"text-color","anchor":"after_divide"}],"connections":[{"to":"after_divide","from":"before_divide","color":"green","label":"divide both sides by 2"}]},"order":3,"title":"Divide by two","description":"Divide both sides by 2. The exact solution earns A1; there is no need to replace e with a decimal."}]}],"generatedAt":"2026-07-20T16:27:19.076Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":75}},{"id":"ea15c29b-d0c6-4762-aef7-bd0945d0dec2","revisionId":"4a963e32-8652-4688-842c-f6ce007ff0f8","questionNumber":"8","questionType":"LA","specification":"Given that $y = \\log_2 x$, find expressions in terms of $y$ for","difficulty":2,"difficultyBand":"easy","totalMarks":6,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"parts":[{"id":"72ca26f5-371e-4ac7-a9ff-8267ccbd2ac5","label":"a","content":"$$\\log_2 x^2$","markscheme":"- Use power law: $2 \\log_2 x$ M1\n- Substitute $y$: $2y$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Applies power law.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $2y$.","code":"A"}],"finalAnswers":["2y"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"power_law","label":"Use the logarithm power law","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\log_2 \\htmlData{anchor=power}{x^2}$"},{"latex":"$$\\htmlData

1.6 Exponentials and Logarithms

Question 6

1.6 Exponentials and Logarithms Questions

1.6 Exponentials and Logarithms

Question 6

1.6 Exponentials and Logarithms Questions