OCR section 1.06 introduces ex — the one function that is its own derivative — its inverse ln x, the laws of logarithms, and the genuinely useful skill of linearising data with logs to test whether a model fits.
The log-linearisation section is the one that appears in real exam data questions. Work through each screen, answer the questions as you go and collect ⭐ stars. Press Start when you're ready.
y = ax is an exponential function. Among all of them, exactly one has gradient equal to its own value at every point — the one with base e = 2.71828…
Misconception: ex is not xe, and its derivative is not xex−1. The power rule does not apply — the variable is in the exponent, not the base.
loga x answers the question ‘what power of a gives x?’. It is the inverse of the exponential: ay = x ⇔ loga x = y. ln x means loge x.
Combine: log₃[x(x − 2)] = 1 ⇒ x(x − 2) = 3¹ = 3.
x² − 2x − 3 = 0 ⇒ (x − 3)(x + 1) = 0 ⇒ x = 3 or x = −1.
Reject x = −1: you cannot take the log of a negative number. ∴ x = 3.
Check: log₃3 + log₃1 = 1 + 0 = 1. ✓
Never lose that mark: a log equation that becomes a quadratic will very often produce a root that must be rejected because it makes a log negative or zero. Always check both.
The variable is stuck in the exponent. Take logs of both sides to bring it down.
Take natural logs: ln(3x) = ln 20 ⇒ x ln 3 = ln 20.
x = ln 20 / ln 3 = 2.99573 / 1.09861 = 2.727 (3 d.p.).
Check: 32.727 ≈ 20.0. ✓ And a sanity check: 3² = 9 and 3³ = 27, so x must lie between 2 and 3. ✓
Misconception: ln 20 / ln 3 is not ln(20/3). Dividing two logs is not the same as the log of a quotient — ln(20/3) = ln 6.67 = 1.897, nowhere near 2.727.
Anything whose rate of change is proportional to its current size follows an exponential model: populations, radioactive decay, cooling, compound interest, drug concentration.
500 e−0.2t = 100 ⇒ e−0.2t = 0.2.
Take ln: −0.2t = ln 0.2 = −1.60944.
t = 1.60944 / 0.2 = 8.05 (3 s.f.).
Check: 500 e−0.2(8.047) = 500 × 0.2 = 100. ✓
Interpreting k: in A = A₀ekt, k is the proportional rate of change per unit time. And always state the model’s limitations — unlimited exponential growth is never realistic for a real population.
Here is the technique OCR examines with real data. If you suspect a power law or an exponential law, take logs — a correct model becomes a straight line.
The whole trick: which one you plot tells you which model you are testing. log–log tests a power law; log y against plain x tests an exponential law.
Comparing with log y = n log x + log a: n = 1.5 and log₁₀a = 0.7.
So a = 100.7 = 5.01 (3 s.f.), and the model is y = 5.01 x1.5.
Tap a logarithm, then tap the value it equals.
Tap an expression on the left, then what it equals.
ex: the function that is its own derivative; d/dx(ekx) = kekx
ln x: the inverse of ex; eln x = x and ln(ex) = x
Log laws: log a + log b = log(ab); log a − log b = log(a/b); n log a = log(an)
Solving ax = b: take logs ⇒ x = ln b / ln a — and this is not ln(b/a)
Modelling: A = A₀ekt; k > 0 growth, k < 0 decay
Linearising: log y vs log x tests y = axn; log y vs x tests y = abx
Reject roots: log equations often give a root that makes a log undefined — check it
That is OCR 1.06 — and the log-linearising skill turns up in real data questions. Press Finish to see your score.
You've worked through Exponentials and logarithms for OCR A-level Mathematics A. 🎉
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