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Logarithms","calculatorUse":null,"isCalculation":true,"parts":[{"id":"e94077ab-6ba2-4581-a28b-3122e83ff7e0","label":"0","content":"Solve the equation, giving your answers in exact form.\n\n$$2e^y + 15e^{-y} = 11$$","markscheme":"- Multiply by $e^y$: $2e^{2y} + 15 = 11e^y$ M1\n- Form quadratic: $2e^{2y} - 11e^y + 15 = 0$ M1\n- Factorise: $(2e^y - 5)(e^y - 3) = 0$ M1\n- Solve for $e^y$: $e^y = \\frac{5}{2}$ or $e^y = 3$ A1\n- Solve for $y$: $y = \\ln\\left(\\frac{5}{2}\\right)$ or $y = \\ln 3$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Multiplies through by $e^y$ to remove negative exponent.","code":"M"},{"id":"p2","marks":1,"description":"Sets up a quadratic equation in $e^y$.","code":"M"},{"id":"p3","marks":1,"description":"Factorises or uses quadratic formula to find values for $e^y$.","code":"M"},{"id":"p4","marks":1,"description":"Identifies $e^y = 2.5$ and $e^y = 3$.","code":"A"},{"id":"p5","marks":1,"description":"Final answers $\\ln(2.5)$ and $\\ln 3$.","code":"A"}],"finalAnswers":["\\ln(2.5), \\ln 3"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"quadratic_substitution","label":"Rewrite as a quadratic","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=original}{\\color{@sky}{2e^y+15e^{-y}=11}}$"},{"latex":"$$\\htmlData{anchor=multiplied}{\\color{@sky}{2e^y e^y+15e^y e^{-y}=11e^y}}$"},{"latex":"$$\\htmlData{anchor=combine}{\\color{@amber}{2e^{2y}+15e^y e^{-y}=11e^y}}$"},{"latex":"$$\\htmlData{anchor=cancel}{\\color{@green}{2e^{2y}+15=11e^y}}$"}],"connections":[{"to":"multiplied","from":"original","color":"sky","label":"multiply every term by e to power y"},{"to":"combine","from":"multiplied","color":"amber","label":"add the indices"},{"to":"cancel","from":"combine","color":"green","label":"reciprocal powers cancel"}]},"order":0,"title":"Clear the negative power","description":"Multiply every term by e raised to the power y. Then use the index laws to combine the positive powers and cancel the reciprocal powers; this earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=before}{\\color{@orange}{2e^{2y}+15=11e^y}}$"},{"latex":"$$\\htmlData{anchor=quadratic}{\\color{@orange}{2e^{2y}-11e^y+15=0}}$"}],"connections":[{"to":"quadratic","from":"before","color":"orange","label":"subtract the right-hand term"}]},"order":1,"title":"Form the quadratic","description":"Subtract the right-hand exponential term from both sides. Writing the equation in standard quadratic form earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=quadratic}{\\color{@green}{2e^{2y}-11e^y+15=0}}$"},{"latex":"$$\\htmlData{anchor=split}{\\color{@green}{2e^{2y}-6e^y-5e^y+15=0}}$"},{"latex":"$$\\htmlData{anchor=grouped}{\\color{@teal}{2e^y(e^y-3)-5(e^y-3)=0}}$"},{"latex":"$$\\htmlData{anchor=factored}{\\color{@green}{(2e^y-5)(e^y-3)=0}}$"}],"connections":[{"to":"split","from":"quadratic","color":"green","label":"split the middle term"},{"to":"grouped","from":"split","color":"teal","label":"group by common factors"},{"to":"factored","from":"grouped","color":"green","label":"take out the common bracket"}]},"order":3,"title":"Factorise the quadratic","description":"To split the middle term, use minus six and minus five: their product is positive thirty and their sum is minus eleven. Splitting and grouping shows where the factors come from; the correct factorisation earns the third M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=first}{\\color{@sky}{2e^y-5=0}}$"},{"latex":"$$\\htmlData{anchor=first_solution}{\\color{@sky}{e^y=\\dfrac{5}{2}}}$"},{"latex":"$$\\htmlData{anchor=second}{\\color{@purple}{e^y-3=0}}$"},{"latex":"$$\\htmlData{anchor=second_solution}{\\color{@purple}{e^y=3}}$"}],"connections":[{"to":"first_solution","from":"first","color":"sky","label":"rearrange"},{"to":"second_solution","from":"second","color":"purple","label":"rearrange"}]},"order":4,"title":"Solve for the exponential","description":"By the zero-product rule, at least one factor must equal zero. Obtaining both exponential values earns A1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=first_exp}{\\color{@teal}{e^y=\\dfrac{5}{2}}}$"},{"latex":"$$\\htmlData{anchor=first_log}{\\color{@teal}{y=\\ln\\!\\left(\\dfrac{5}{2}\\right)}}$"},{"latex":"$$\\htmlData{anchor=second_exp}{\\color{@indigo}{e^y=3}}$"},{"latex":"$$\\htmlData{anchor=second_log}{\\color{@indigo}{y=\\ln 3}}$"}],"connections":[{"to":"first_log","from":"first_exp","color":"teal","label":"take natural logarithms"},{"to":"second_log","from":"second_exp","color":"indigo","label":"take natural logarithms"}]},"order":5,"title":"Take natural logarithms","description":"The natural logarithm reverses the exponential function. Apply it to each branch to obtain the exact answers and earn the final A1 mark."}]}],"generatedAt":"2026-07-20T16:29:54.676Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":53},"walkthroughVideo":null},{"id":"c6a2be8b-037e-4010-ac66-2e0358c65a4f","revisionId":"2ce60153-8e9e-4acc-ae06-aa9c82eadcf2","questionNumber":"9","questionType":"LA","specification":"","difficultyBand":"medium","tier":null,"totalMarks":5,"sourceTitle":"Exponentials and Logarithms","calculatorUse":null,"isCalculation":true,"parts":[{"id":"e08020a6-cbbb-4295-9b5b-7b15de10018f","label":"0","content":"Solve the equation, giving your answers in exact form.\n\n$$2e^y + 15e^{-y} = 11$$","markscheme":"- Multiply by $e^y$: $2e^{2y} + 15 = 11e^y$ M1\n- Form quadratic: $2e^{2y} - 11e^y + 15 = 0$ M1\n- Factorise: $(2e^y - 5)(e^y - 3) = 0$ M1\n- Solve for $e^y$: $e^y = \\frac{5}{2}$ or $e^y = 3$ A1\n- Solve for $y$: $y = \\ln\\left(\\frac{5}{2}\\right)$ or $y = \\ln 3$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Multiplies through by $e^y$ to eliminate the negative exponent.","code":"M"},{"id":"p2","marks":1,"description":"Rearranges into a quadratic form in $e^y$.","code":"M"},{"id":"p3","marks":1,"description":"Factorises or uses the quadratic formula to find values for $e^y$.","code":"M"},{"id":"p4","marks":1,"description":"Identifies $e^y = 2.5$ and $e^y = 3$.","code":"A"},{"id":"p5","marks":1,"description":"Finds exact values for $y$ using natural logarithms.","code":"A"}],"finalAnswers":["\\ln(2.5), \\ln(3)"],"generatedAt":"2026-08-27T05:50:41.631Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"516d0a9c-9e77-486d-97a5-47044f123d7b","revisionId":"716818a0-23ef-4ae9-b1d7-3c910da5bca8","questionNumber":"10","questionType":"LA","specification":"The population of a species of plant in a field is modelled using the formula $P = 50e^{0.1t}$\nWhere $t$ is the number of weeks since the population was first recorded.","difficultyBand":"medium","tier":null,"totalMarks":7,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"isCalculation":true,"parts":[{"id":"933900be-a9cc-47c9-ad6a-a913c57e59ae","label":"a","content":"Write down the number of the plants when the population was first recorded.","markscheme":"- $t=0 \\Rightarrow P = 50e^0 = 50$ B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"States 50.","code":"B"}],"finalAnswers":["50"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"substitute_initial_time","label":"Substitute the initial time","steps":[{"type":"text","order":0,"title":"Identify initial time","content":"The phrase “when the population was first recorded” means the elapsed time is $t=0$.","highlights":[{"text":"when the population was first recorded","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$P=50e^{\\htmlData{anchor=exponent_before}{\\color{@sky}{0.1(0)}}}$"},{"latex":"$$P=50e^{\\htmlData{anchor=exponent_after}{\\color{@sky}{0}}}$"},{"latex":"$$P=50\\times1$"},{"latex":"$$P=\\boxed{50\\text{ plants}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"exponent_before"},{"color":"sky","style":"text-color-with-background","anchor":"exponent_after"}],"connections":[{"to":"exponent_after","from":"exponent_before","color":"sky","label":"simplify the exponent"}]},"order":1,"title":"Substitute and evaluate","description":"Put t = 0 into the model, then use e⁰ = 1. Obtaining the initial population P = 50 earns B1."}]}],"generatedAt":"2026-07-20T16:21:53.884Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"6981d984-e457-4d1c-b969-97969a861a4f","label":"b","content":"Find the rate of increase in the population 10 weeks after the population was first recorded.","markscheme":"- Differentiate: $\\frac{dP}{dt} = 5e^{0.1t}$ M1\n- Substitute $t=10$: $5e^{0.1(10)} = 5e$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Differentiates $P$ with respect to $t$.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $5e$.","code":"A"}],"finalAnswers":["5e"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"differentiate_then_substitute","label":"Differentiate then substitute","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$P=50e^{0.1t}$"},{"latex":"$$\\frac{dP}{dt}=\\color{@orange}{\\htmlData{anchor=coefficient}{50(0.1)}}\\color{@sky}{\\htmlData{anchor=exponential}{e^{0.1t}}}$"},{"latex":"$$\\frac{dP}{dt}=\\color{@sky}{\\htmlData{anchor=derivative}{5e^{0.1t}}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"coefficient"},{"color":"sky","style":"text-color","anchor":"derivative"}]},"order":1,"title":"Write the derivative","description":"Differentiate with respect to t. Since the derivative of e^(0.1t) is 0.1e^(0.1t), the coefficient becomes 50 × 0.1 = 5. M1: dP/dt = 5e^(0.1t)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{dP}{dt}=5e^{0.1t}$"},{"latex":"$$\\left.\\frac{dP}{dt}\\right|_{t=10}=\\color{@orange}{\\htmlData{anchor=substitution}{5e^{0.1(10)}}}$"},{"latex":"$$\\left.\\frac{dP}{dt}\\right|_{t=10}=\\htmlData{anchor=power}{5e^1}$"},{"latex":"$$\\left.\\frac{dP}{dt}\\right|_{t=10}=\\color{@green}{\\htmlData{anchor=answer}{5e\\text{ plants per week}}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"substitution"},{"color":"green","style":"text-color-with-background","anchor":"answer"}]},"order":2,"title":"Substitute ten weeks","description":"Evaluate the derivative at the required time. A1: substitute t = 10 and simplify to 5e."}]}],"generatedAt":"2026-07-20T16:29:34.276Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"238260fe-cf0f-44a6-9ddc-ce5da53bf72d","label":"c","content":"Find how many weeks it takes for the number of plants to exceed 300.","markscheme":"- Set $P=300$: $300 = 50e^{0.1t}$ M1\n- $6 = e^{0.1t}$ M1\n- $\\ln 6 = 0.1t \\Rightarrow t = 10 \\ln 6$ M1\n- $t \\approx 17.9$, so 18 weeks A1","marks":4,"order":2,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Sets up equation $300 = 50e^{0.1t}$.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to $e^{0.1t} = 6$.","code":"M"},{"id":"p3","marks":1,"description":"Uses logs to solve for $t$.","code":"M"},{"id":"p4","marks":1,"description":"Final answer 18 weeks.","code":"A"}],"finalAnswers":["18"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"natural_logarithms","label":"Solve using natural logarithms","steps":[{"type":"text","order":0,"title":"Set the threshold","content":"To find when the population starts to exceed $300$, first find the boundary time when it is exactly $300$. Substitute $P=300$ into the model:\n\n$$300=50e^{0.1t}$$\n\nThis correct equation earns the first **M1** mark.","highlights":[{"text":"exceed 300","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$300=\\htmlData{anchor=coefficient_start}{\\color{@orange}{50}}e^{0.1t}$"},{"latex":"$$\\htmlData{anchor=whole_quotient}{\\dfrac{300}{\\htmlData{anchor=coefficient_divisor}{\\color{@orange}{50}}}}=e^{0.1t}$"},{"latex":"$$\\htmlData{anchor=six}{\\color{@sky}{6}}=e^{0.1t}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"coefficient_start"},{"color":"orange","style":"text-color","anchor":"coefficient_divisor"},{"color":"sky","style":"text-color","anchor":"six"}],"connections":[{"to":"coefficient_divisor","from":"coefficient_start","color":"orange","label":"divide both sides by 50"},{"to":"six","from":"whole_quotient","color":"sky","label":"simplify"}]},"order":1,"title":"Isolate the exponential","description":"Divide both sides by 50. Reaching 6 = e^(0.1t) earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$6=e^{\\htmlData{anchor=exponent}{\\color{@orange}{0.1t}}}$"},{"latex":"$$\\ln 6=\\htmlData{anchor=linear_equation}{\\color{@orange}{0.1t}}$"},{"latex":"$$\\htmlData{anchor=divided_equation}{\\color{@sky}{t=\\htmlData{anchor=quotient}{\\dfrac{\\ln 6}{0.1}}}}$"},{"latex":"$$t=\\htmlData{anchor=product}{\\color{@green}{10\\ln 6}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"exponent"},{"color":"sky","style":"text-color","anchor":"divided_equation"},{"color":"green","style":"text-color","anchor":"product"}],"connections":[{"to":"linear_equation","from":"exponent","color":"orange","label":"use ln(e^x) = x"},{"to":"divided_equation","from":"linear_equation","color":"sky","label":"divide by 0.1"},{"to":"product","from":"quotient","color":"green","label":"since 1 ÷ 0.1 = 10"}]},"order":2,"title":"Take natural logarithms","description":"Take natural logarithms of both sides. Using ln(e^x) = x removes the exponential; obtaining t = 10 ln 6 earns the third M1 mark."},{"type":"text","order":3,"title":"Round to whole weeks","content":"Evaluate the exact result:\n\n$$t=10\\ln 6\\approx17.9$$\n\nThe population reaches $300$ after about $17.9$ weeks. The question asks for the number of **whole weeks** until it exceeds $300$, so round **up**:\n\n$$\\boxed{18\\text{ weeks}}$$\n\nThe accurate calculation and correct interpretation earn **A1**.","highlights":[{"text":"how many weeks","location":"specification"},{"text":"exceed 300","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:39:07.858Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":47},"walkthroughVideo":null}],"subjectCode":"as_level_maths","summaries":[{"id":"5e711279-248d-4041-8b5b-944dc2a079e2","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"71ec8bdb-d5b0-433a-9316-f3aafebdd276","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"66787241-20fe-4677-a28d-e2080738231a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"ac0c508c-786a-4c5a-b076-8d1fca860f98","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"9637e647-82d8-412c-ab92-13583c6f5d65","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e5a61db3-1c05-4d3f-982e-8cb54f173814","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"0504f7aa-8e4b-4c41-99aa-41eab8797491","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"100baeb7-11ee-44ce-8593-abbac07096d8","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c1524aa2-f8bc-4f32-89b3-4ab29cc37f0f","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f41109a7-42e0-417b-8d30-f489b624e4a7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e4f7a0f1-0bc3-4b5c-8cd1-e6acc6abbaa6","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"feae588b-ccc6-49d9-910b-e2c20ede5492","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"297a6c62-3939-4bf2-b695-5abc3e67f5c7","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"88185f03-c7d9-4d3b-8a62-0dec9d95df5f","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a58044a8-a4d6-48ff-96df-48be4f3ec4d1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"ea15c29b-d0c6-4762-aef7-bd0945d0dec2","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9434c3ed-c63c-4953-811c-6170426c7815","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"c6a2be8b-037e-4010-ac66-2e0358c65a4f","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"516d0a9c-9e77-486d-97a5-47044f123d7b","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"54a4c1e3-f02f-480a-9b20-c0b1f667e828","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"64ea9803-5740-4af1-9e60-de02022423cb","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fd00b03b-1e6d-4862-906a-adce6e680d80","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"3fb0edd5-94e8-4403-8a78-34fe1055baf5","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"9d18c848-dc6f-45d9-b2ae-030c82d05813","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"4f882717-8c90-4f68-af94-914e8330f88f","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"c213fe79-05e5-4985-9497-74ee6ceed769","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"afdf1b4b-d6cb-43ac-9c97-67360967909c","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"793c92a1-46cb-4a81-a3e6-121c2bbdfb57","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"efb9f39e-5783-4f3d-b0ff-8cb66a8a15f6","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"a0b6a559-f180-4915-9c95-f8c861f5c721","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"29663621-97bc-41a3-a277-f410fef439c6","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"1bfcb2ae-52cb-4aca-9e4c-af12063044cd","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType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and Logarithms","topicId":"b381dbdf-209e-44e8-95f7-830172c37463","topicSidebar":"$L3a","topicSlug":"as-pure-mathematics-14-exponentials-and-logarithms","topicTitle":"Exponentials and Logarithms"}]}],"$L3b","$L3c"]}]

Exponentials and Logarithms

Question 6

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 6

Exponentials and Logarithms Questions