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AS LEVEL MATHS EXPONENTIAL AND LOGS, Exercises of Mathematics

Mathematics

AS LEVEL MATHS EXPONENTIAL AND LOGS

Typology: Exercises

2025/2026

Uploaded on 06/04/2026

jk008
jk008 ๐Ÿ‡ฌ๐Ÿ‡ง

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EXPONENTIALS & LOGARITHMS
Pure Year 1, Chapter 14
Exponentials
Solve 4๐‘ฅ= 7
Solve 2๐‘ฅ= 32๐‘ฅ+1 giving your answer in the form ๐‘Žln 3
๐‘ln 2+๐‘ ln 3
Solve 5๐‘ฅ+1 = 52๐‘ฅ โˆ’ 6 giving your answer(s) to 3dp
Differentiate ๐‘ฆ = 4๐‘’โˆ’2๐‘ฅ with respect to ๐‘ฅ
Graphs
Differentiating Exponentials
If ๐‘ฆ = ๐‘’๐‘ฅ then ๐‘‘๐‘ฆ
๐‘‘๐‘ฅ =
If ๐‘ฆ = ๐‘’๐‘˜๐‘ฅ then ๐‘‘๐‘ฆ
๐‘‘๐‘ฅ =
(Year 2: If ๐‘ฆ = ๐‘Ž๐‘ฅ then ๐‘‘๐‘ฆ
๐‘‘๐‘ฅ =ln ๐‘Ž ร— ๐‘Ž๐‘ฅ)
Solving Equations
Log Laws
log ๐‘Ž + log ๐‘ =
log ๐‘Ž โˆ’ log ๐‘ =
log ๐‘Ž๐‘=
Remember that you cannot input a negative into a
logarithm โ€“ check your answers to see if they are valid
Look out for pseudo-quadratics with exponentials!
pf3
pf4

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EXPONENTIALS & LOGARITHMS

Pure Year 1 , Chapter 14

Exponentials

Solve 4 ๐‘ฅ^ = 7

Solve 2 ๐‘ฅ^ = 32 ๐‘ฅ+^1 giving your answer in the form

๐‘Ž ln 3 ๐‘ ln 2 +๐‘ ln 3

Solve 5

โˆ’ 6 giving your answer(s) to 3 dp

Differentiate ๐‘ฆ = 4 ๐‘’โˆ’^2 ๐‘ฅ^ with respect to ๐‘ฅ

Graphs Differentiating Exponentials If ๐‘ฆ = ๐‘’๐‘ฅ^ then ๐‘‘๐‘ฆ ๐‘‘๐‘ฅ = If ๐‘ฆ = ๐‘’๐‘˜๐‘ฅ^ then ๐‘‘๐‘ฆ ๐‘‘๐‘ฅ = (Year 2: If ๐‘ฆ = ๐‘Ž๐‘ฅ^ then ๐‘‘๐‘ฆ ๐‘‘๐‘ฅ =^ ln^ ๐‘Ž^ ร—^ ๐‘Ž

Solving Equations Log Laws log ๐‘Ž + log ๐‘ = log ๐‘Ž โˆ’ log ๐‘ = log ๐‘Ž๐‘^ = Remember that you cannot input a negative into a logarithm โ€“ check your answers to see if they are valid Look out for pseudo-quadratics with exponentials!

Solving Equations with Logarithms

Solve 2 log 2 ๐‘ฅ โˆ’ log 2 3 = 5

Given that ๐ด = 3 ๐‘’

, find the value of ๐‘ก when ๐ด = 13. 5

Solve 2 ln(๐‘ฅ โˆ’ 2 ) + ln 4 = 2 ln(๐‘ฅ + 1 )

Logarithmic and Non-Linear Data on Graphs The value, ๐‘‰ pounds, of a printer, ๐‘ก years after it was bought is modelled by ๐‘‰ = ๐‘Ž๐‘๐‘ก where ๐‘Ž and ๐‘ are constants. When log 10 ๐‘‰ is plotted against ๐‘ก, the relationship is linear, and the line passes through the points 0 , 2. 5 and ( 7 , 2 ) a) Using these points, find a complete equation for ๐‘‰ in terms of ๐‘ก, giving ๐‘Ž and ๐‘ to 3 significant figures. b) Exactly 5 years after the printer was bought, the value of the printer was ยฃ 100. Use this information to evaluate the reliability of the model Logarithms can be used to change from exponential/non-linear to linear data. We can then use the maths of straight line graphs to solve problems.

  1. Take logs
  2. Find the equation of the line if necessary
  3. Compare both equations
  4. Solve for the unknowns

Exam Tip : if a model is within 10 % of the observed value, we can say it

is a good/reliable model โ€“ otherwise, we say it is not good/unreliable