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2}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"power_form"},{"color":"amber","style":"text-color","anchor":"evaluated"}]},"order":1,"title":"Move the coefficient","description":"Use the power law: move the coefficient inside the logarithm as a power. This earns the method mark M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=difference}{\\log_a 64-\\log_a 2}$"},{"latex":"$$\\htmlData{anchor=quotient}{\\log_a\\!\\left(\\dfrac{64}{2}\\right)}$"},{"latex":"$$\\htmlData{anchor=answer}{\\log_a 32}$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"difference"},{"color":"teal","style":"text-color","anchor":"quotient"},{"color":"green","style":"text-color-with-background","anchor":"answer"}]},"order":2,"title":"Combine the logarithms","description":"Use the quotient law: subtraction of logarithms becomes division inside one logarithm. Simplifying correctly earns the accuracy mark A1."}]}],"generatedAt":"2026-07-20T16:42:27.169Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":80},"walkthroughVideo":null},{"id":"9d9d38e1-ea87-493e-9654-ae3f321fb6af","revisionId":"837fd8bf-c267-4d2e-8b58-d8c4fac50599","questionNumber":"39","questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"isCalculation":true,"parts":[{"id":"d9e45578-f8c9-49b3-800e-b1ecf1dbf892","label":"0","content":"The value of an investment grew exponentially from £3.2 million in 2001 to £6.1 million in 2021. Estimate the value of the investment in 2031 if the exponential growth continued.","markscheme":"- $V = 3.2e^{kt}$. At $t=20, V=6.1$: $6.1 = 3.2e^{20k}$ M1\n- $e^{20k} = \\frac{6.1}{3.2} \\Rightarrow k = 0.0323$ M1\n- At $t=30, V = 3.2e^{0.0323(30)} = 8.4$ million A1","marks":3,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Finds the growth constant $k$ or growth factor.","code":"M"},{"id":"p2","marks":1,"description":"Final answer 8.4 million.","code":"A"}],"finalAnswers":["8.4"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"exponential-model","label":"Build an exponential model","steps":[{"type":"text","order":0,"title":"Set the time origin","content":"Let $t$ be the number of years after 2001, and let $V$ be the investment value in millions of pounds. Exponential growth has the model\n\n$$V=3.2e^{kt},$$\n\nwhere $k$ is a constant growth rate. This uses the starting value when $t=0$.\n\nThe phrase “from £3.2 million in 2001” therefore gives the $3.2$ in the model.","highlights":[{"text":"grew exponentially from £3.2 million in 2001","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"6.1=3.2e^{20\\htmlData{anchor=k_a}{k}}"},{"latex":"e^{20\\htmlData{anchor=k_b}{k}}=\\frac{6.1}{3.2}"},{"latex":"20k=\\ln\\!\\left(\\frac{6.1}{3.2}\\right)"},{"latex":"\\htmlData{anchor=k_d}{k}=0.0323"}],"highlights":[{"color":"green","style":"box","anchor":"k_d"}],"connections":[{"to":"k_b","from":"k_a","color":"sky","label":"divide by 3.2"}]},"order":1,"title":"Find the growth rate","description":"Since 2021 is 20 years after 2001, use $t=20$ and $V=6.1$ to form the first equation; this earns the first method mark. Taking logarithms and obtaining $k=0.0323$ earns the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"V=3.2e^{\\htmlData{anchor=exp_a}{kt}}"},{"latex":"V=3.2e^{\\htmlData{anchor=exp_b}{0.0323(30)}}"},{"latex":"\\boxed{V=\\htmlData{anchor=value_a}{8.4}\\text{ million}}"}],"connections":[{"to":"exp_b","from":"exp_a","color":"purple","label":"substitute k and t"},{"to":"value_a","from":"exp_b","color":"green","label":"evaluate"}]},"order":2,"title":"Predict the 2031 value","description":"The year 2031 is 30 years after 2001, so substitute $t=30$ and $k=0.0323$. The final value is in millions of pounds, and this earns the accuracy mark."}]},{"id":"growth-factor","label":"Scale the growth factor","steps":[{"type":"text","order":0,"title":"Find the growth factor","content":"Over the $20$ years from $2001$ to $2021$, the investment is multiplied by\n\n$$\\frac{6.1}{3.2}=1.90625.$$\n\nSo the $20$-year growth factor is $1.90625$. Equal time intervals in exponential growth have equal multiplication factors.","highlights":[{"text":"from £3.2 million in 2001 to £6.1 million in 2021","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"6.1=3.2e^{\\htmlData{anchor=twenty_k}{20k}}"},{"latex":"e^{20k}=\\htmlData{anchor=factor_ratio}{\\frac{6.1}{3.2}}"},{"latex":"\\htmlData{anchor=logged_exponent}{20k}=\\ln\\left(\\frac{6.1}{3.2}\\right)"},{"latex":"k=\\frac{\\ln\\left(\\frac{6.1}{3.2}\\right)}{20}=\\htmlData{anchor=rate_k}{0.0323}"}],"connections":[{"to":"factor_ratio","from":"twenty_k","color":"sky","label":"divide by 3.2"},{"to":"logged_exponent","from":"factor_ratio","color":"purple","label":"take logs"},{"to":"rate_k","from":"logged_exponent","color":"green","label":"divide by 20"}]},"order":1,"title":"Confirm the rate","description":"This setup gains the first method mark. Isolating the factor and then finding the rate gains the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{\\htmlData{anchor=years_to_2031}{30}}{\\htmlData{anchor=known_years}{20}}=\\htmlData{anchor=intervals}{1.5}"},{"latex":"\\text{growth factor for 30 years}=\\left(1.90625\\right)^{\\htmlData{anchor=power_30}{1.5}}"}],"connections":[{"to":"power_30","from":"intervals","color":"purple","label":"use as power"}]},"order":2,"title":"Scale to thirty years","description":"Thirty years is 30 divided by 20 known intervals. Raise the 20-year factor to this power."},{"type":"hyper_latex","block":{"nodes":[{"latex":"V=3.2\\left(1.90625\\right)^{1.5}"},{"latex":"V=8.4220877756\\text{ million}"},{"latex":"\\boxed{V\\approx\\htmlData{anchor=final_value}{8.4}\\text{ million}}"}],"highlights":[{"color":"amber","style":"box","anchor":"final_value"}]},"order":3,"title":"Calculate the estimate","description":"Multiply the initial value by this 30-year growth factor, then round the estimate to 2 significant figures."}]}],"generatedAt":"2026-09-01T09:18:59.763Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"d9e45578-f8c9-49b3-800e-b1ecf1dbf892:exponential-model","label":"Build an exponential model","steps":[{"type":"text","order":0,"title":"Set the time origin","content":"Let $t$ be the number of years after 2001, and let $V$ be the investment value in millions of pounds. Exponential growth has the model\n\n$$V=3.2e^{kt},$$\n\nwhere $k$ is a constant growth rate. This uses the starting value when $t=0$.\n\nThe phrase “from £3.2 million in 2001” therefore gives the $3.2$ in the model.","highlights":[{"text":"grew exponentially from £3.2 million in 2001","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"6.1=3.2e^{20\\htmlData{anchor=k_a}{k}}"},{"latex":"e^{20\\htmlData{anchor=k_b}{k}}=\\frac{6.1}{3.2}"},{"latex":"20k=\\ln\\!\\left(\\frac{6.1}{3.2}\\right)"},{"latex":"\\htmlData{anchor=k_d}{k}=0.0323"}],"highlights":[{"color":"green","style":"box","anchor":"k_d"}],"connections":[{"to":"k_b","from":"k_a","color":"sky","label":"divide by 3.2"}]},"order":1,"title":"Find the growth rate","description":"Since 2021 is 20 years after 2001, use $t=20$ and $V=6.1$ to form the first equation; this earns the first method mark. Taking logarithms and obtaining $k=0.0323$ earns the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"V=3.2e^{\\htmlData{anchor=exp_a}{kt}}"},{"latex":"V=3.2e^{\\htmlData{anchor=exp_b}{0.0323(30)}}"},{"latex":"\\boxed{V=\\htmlData{anchor=value_a}{8.4}\\text{ million}}"}],"connections":[{"to":"exp_b","from":"exp_a","color":"purple","label":"substitute k and t"},{"to":"value_a","from":"exp_b","color":"green","label":"evaluate"}]},"order":2,"title":"Predict the 2031 value","description":"The year 2031 is 30 years after 2001, so substitute $t=30$ and $k=0.0323$. The final value is in millions of pounds, and this earns the accuracy mark."}]},{"id":"d9e45578-f8c9-49b3-800e-b1ecf1dbf892:growth-factor","label":"Scale the growth factor","steps":[{"type":"text","order":0,"title":"Find the growth factor","content":"Over the $20$ years from $2001$ to $2021$, the investment is multiplied by\n\n$$\\frac{6.1}{3.2}=1.90625.$$\n\nSo the $20$-year growth factor is $1.90625$. Equal time intervals in exponential growth have equal multiplication factors.","highlights":[{"text":"from £3.2 million in 2001 to £6.1 million in 2021","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"6.1=3.2e^{\\htmlData{anchor=twenty_k}{20k}}"},{"latex":"e^{20k}=\\htmlData{anchor=factor_ratio}{\\frac{6.1}{3.2}}"},{"latex":"\\htmlData{anchor=logged_exponent}{20k}=\\ln\\left(\\frac{6.1}{3.2}\\right)"},{"latex":"k=\\frac{\\ln\\left(\\frac{6.1}{3.2}\\right)}{20}=\\htmlData{anchor=rate_k}{0.0323}"}],"connections":[{"to":"factor_ratio","from":"twenty_k","color":"sky","label":"divide by 3.2"},{"to":"logged_exponent","from":"factor_ratio","color":"purple","label":"take logs"},{"to":"rate_k","from":"logged_exponent","color":"green","label":"divide by 20"}]},"order":1,"title":"Confirm the rate","description":"This setup gains the first method mark. Isolating the factor and then finding the rate gains the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{\\htmlData{anchor=years_to_2031}{30}}{\\htmlData{anchor=known_years}{20}}=\\htmlData{anchor=intervals}{1.5}"},{"latex":"\\text{growth factor for 30 years}=\\left(1.90625\\right)^{\\htmlData{anchor=power_30}{1.5}}"}],"connections":[{"to":"power_30","from":"intervals","color":"purple","label":"use as power"}]},"order":2,"title":"Scale to thirty years","description":"Thirty years is 30 divided by 20 known intervals. Raise the 20-year factor to this power."},{"type":"hyper_latex","block":{"nodes":[{"latex":"V=3.2\\left(1.90625\\right)^{1.5}"},{"latex":"V=8.4220877756\\text{ million}"},{"latex":"\\boxed{V\\approx\\htmlData{anchor=final_value}{8.4}\\text{ million}}"}],"highlights":[{"color":"amber","style":"box","anchor":"final_value"}]},"order":3,"title":"Calculate the estimate","description":"Multiply the initial value by this 30-year growth factor, then round the estimate to 2 significant figures."}]}],"generatedAt":"2026-09-01T09:19:17.829Z","modelVersion":"gpt-5.6-sol+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"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":"LA","isCalculation":true},{"id":"ec9dd2f1-a8eb-4350-9d79-abce80fcda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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 20

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 20

Exponentials and Logarithms Questions