Knowledge organisers

Exponentials and logarithms: A Level Maths knowledge organiser

Everything to know about exponentials and logarithms on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Exponential function: y = aˣ with a > 0; y = eˣ is the one whose gradient equals its value.
Natural logarithm: ln x = logₑ x, the inverse of eˣ.
Laws of logarithms: log ab = log a + log b, log(a/b) = log a − log b, log aᵏ = k log a.
Linearising: Taking logs of y = axⁿ or y = kbˣ gives a straight line you can read a, n, k, b from.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Definition	\(a^x=b\iff x=\log_ab\)
Laws	\(\log_axy=\log_ax+\log_ay,\) \(\log_a\tfrac xy=\log_ax-\log_ay,\) \(\log_ax^k=k\log_ax\)
Change of baseIn the formula booklet: Edexcel	\(\log_ax=\frac{\log_bx}{\log_ba}\)
Exponential formIn the formula booklet: Edexcel	\(a^x=e^{x\ln a}\)
Inverses	\(e^{\ln x}=x,\) \(\ln e^x=x\)
Linearising	\(y=ax^n\Rightarrow\log y=\log a+n\log x;\) \(y=kb^x\Rightarrow\log y=\log k+x\log b\)

Worked example

Solve 3ˣ = 20, giving x to 3 significant figures.

Take logs: x ln 3 = ln 20
x = ln 20 / ln 3 = 2.7268…

Answer: x = 2.73 (3 s.f.)

Common mistakes

Modelling: model not written out, limitation not in context
Gradient of e^(kx) given as e^(kx)
Initial value read as the growth rate
Log_a(1) = 1

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Sketch graphs of exponential functions y = aˣ for different values of a, describing their intercepts, asymptotes and whether they show growth or decay.
Sketch y = eˣ and its transformations, and use the fact that the gradient of y = e^(kx) is ke^(kx) to describe rates of change.
Understand logarithms as the inverse of exponentials, convert between forms such as 2⁵ = 32 and log₂ 32 = 5, and evaluate simple logs.
Use the laws of logarithms for products, quotients and powers to simplify expressions and solve equations such as log₃ x + log₃ (x − 2) = 1.
Use natural logarithms to solve equations involving eˣ and use log graphs to turn relationships such as y = axⁿ and y = abˣ into straight lines.
Use logarithms to change exponential relationships such as y = abˣ and y = axⁿ into linear form, and interpret regression lines of log data.

The printable sheet

Exponentials and logarithms knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Exponentials and logarithms knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

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