Example 1: Doubling time in a growth model
A population is modelled by P = 200e⁰·⁰⁵ᵗ, where t is in years. Find the time for the population to double.
400 = 200e⁰·⁰⁵ᵗe⁰·⁰⁵ᵗ = 20.05t = ln 2t = ln 2 / 0.05 ≈ 13.86
Answer: About 13.9 years
Exponential and Logarithmic Functions (DSE M1): e, ln x and Linear Relations
e ≈ 2.71828 is the limit of (1 + 1/n)ⁿ, and ln x is the logarithm to base e. M1 uses them for growth and decay models such as P = P₀ekt, and for turning y = kaˣ into the linear relation ln y = ln k + x ln a.
Exponential and Logarithm Formulas (DSE M1)
Definition of e
e = limn→∞ (1 + 1/n)ⁿExponential series
eˣ = 1 + x + x²/2! + x³/3! + …Inverse pair
ln eˣ = x, eln x = x, aˣ = ex ln aGrowth / decay
f(t) = P₀ekt (k > 0), f(t) = P₀e−kt (k > 0)Linearising
y = kaˣln y = ln k + x ln a: slope ln a, intercept ln kLinearising
y = kxⁿln y = ln k + n ln x: slope n, intercept ln kWorked Examples — Exponential and Logarithmic Functions
Example 2: Finding constants from a straight-line graph
Variables x and y are related by y = kaˣ. The graph of ln y against x is a straight line with slope 0.588 and vertical intercept 1.10. Find k and a.
ln y = ln k + x ln a- Slope:
ln a = 0.588, soa = e⁰·⁵⁸⁸ ≈ 1.80 - Intercept:
ln k = 1.10, sok = e¹·¹⁰ ≈ 3.00
Answer: k ≈ 3.00, a ≈ 1.80
💡 Exam Traps & Secrets
⚠️ The intercept is ln k, not kOn a graph of
ln y against x, the intercept gives ln k and the slope gives ln a. Both need e to the power of the reading to recover k and a.⚠️ Which variables go on the axesFor
y = kaˣ plot ln y against x. For y = kxⁿ plot ln y against ln x. Mixing them up gives a curve instead of a line.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Decay)A substance decays as M = 80e^(−0.12t). Find its half-life and show the decay curve.
(Linear relation)y = k xⁿ. The graph of ln y against ln x has slope 1.5 and intercept 0.69. Find k and n.
(Exponential series)Expand e^(−2x) as far as the term in x³.
Frequently Asked Questions
Take natural logarithms: ln y = ln k + x ln a. Plotting ln y against x gives a line with slope ln a and intercept ln k.
eˣ = 1 + x + x²/2! + x³/3! + …. Replacing x by kx gives the series for e to the power kx, which M1 combines with binomial expansions.
Because the rate of change of e to the power kt is k times itself, which is exactly what "growing at a rate proportional to its size" means.