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Exponentials and Logarithms

Powers and their inverse: the rules that quietly drive sequences, financial maths, and calculus across Paper One.

Topic Importance

Exponentials and logarithms has appeared in 23 questions over the last 12 years of past papers. But as we saw in the Paper One overview, the headline number understates it — this is one of the most connected topics on the course.

Exponentials and logarithms appeared in 23 questions over the last 12 years, plus all the questions it is hidden inside

Exponentials turn up across the whole paper, often without being labelled. Sequences and series are built on powers, calculus questions hand you exponential functions to differentiate or integrate, and financial maths writes interest as an exponential.

Exponentials hidden inside other topics: sequences and series use ar to the n, differentiation works on e to the 3x, financial maths uses P times 1 plus i to the power of t

Master the rules here and they keep paying out across the rest of Paper One.

What the Topic Covers

The topic splits into two halves: the rules of exponentials, and then logarithms — the same ideas run in reverse.

Exponentials start with the definition — repeated multiplication written as a power — and build up the rules for combining them:

Definition \(a \times a \times a = a^3\)

Multiplying \(x^3 \times x^2 = x^5\)

Dividing \(\dfrac{x^5}{x^2} = x^3\)

Power of a power \((x^3)^2 = x^6\)

Negative powers \(x^{-n} = \dfrac{1}{x^n}\)

Logarithms begin with what a log even means. There are two ways to see the definition: \(\log_a b\) is the power of \(a\) you need to get \(b\) — so raising \(a\) to it gives \(b\) back. And applying \(\log_a\) to a power of \(a\) just reads off that power.

Definition \(a^{\log_a b} = b\)

Definition \(\log_a(a^b) = b\)

Adding \(\log_a b + \log_a c = \log_a(bc)\)

Subtracting \(\log_a b - \log_a c = \log_a\!\left(\dfrac{b}{c}\right)\)

Extracting powers \(\log_a(b^c) = c\,\log_a b\)

Change of base \(\log_a b = \dfrac{\log_c b}{\log_c a}\)

Exponentials and logarithms are inverses: e to the log base e of x equals x, and log base e of e to the x equals x

Example question

Evaluate \(\log_3 81\).

Exam Correlations

Exponentials and logarithms appears 23 times over 12 years, and its most frequent companions tell a story. The top shared appearances are Algebra 1, Algebra 2, and differentiation — at 9 each.

Questions shared with exponentials and logarithms over 12 years: Algebra 1, Algebra 2, and differentiation at 9 each, then functions 6, integration 4, length area and volume 4, patterns 3, sequences and series 3, algebra 3 and induction 1 each

Calculus is a standout partner: when an exam question involves exponentials or logs, it very often asks you to differentiate them. And the algebra pairing works the same way it did in the algebra articles — an exponential equation usually needs algebraic manipulation to solve.

The usual caveat applies: appearing together does not always mean the mathematics is deeply linked. But the practical point stands — if you cannot handle the exponential and log rules fluently, marks in calculus and algebra questions are at risk, not just the marks labelled "logs".

Concept Connections

Exponentials and logarithms sits right at the start of the hierarchy — it has no real prerequisites. The connections all flow outwards, and they are some of the strongest on the course.

Exponentials and logarithms has no prerequisites and feeds outward into financial maths, sequences and series, calculus, and Algebra 2

Topic A question you will meet there The exponentials idea inside it

Financial Maths \(P(1+i)^t\) Compound interest — the biggest connection. The principal grows by a factor raised to a power: a plain exponential wearing a business suit.

Sequences and Series \(a r^n\) Geometric sequences — every geometric term is a starting value times a power of the ratio.

Calculus \(\dfrac{d}{dx}\, e^{3x}, \quad \displaystyle\int \ln x \, dx\) Exponential and log functions — calculus questions hand you these functions to differentiate or integrate; the rules here are the prerequisite knowledge.

Algebra 2 \(3x^2 + 2x - 8\) Polynomials — a polynomial is just a sum of powers of \(x\) with coefficients. Simple exponentials are the input.

Study Order

The study order follows the same sequence as the topic itself: exponentials first, then logs, with each rule building on the ones before it. Click each step to see how.

Exponential Definition

Everything starts with what a power means: repeated multiplication, written compactly.

\(a \times a \times a = a^3\)

Multiplying Exponentials

Multiplying powers of the same base means adding the powers — expand each one and count: \(x^3 \times x^2\) is three \(x\)s times two \(x\)s, which is five \(x\)s altogether.

\(x^3 \times x^2 = x^5\)

Dividing Exponentials

The obvious next step, because it is just the reverse: dividing powers of the same base subtracts the powers.

\(\dfrac{x^5}{x^2} = x^3\)

Powers of Powers

Expand the outer power using the definition: \((x^3)^2\) means \(x^3 \times x^3\), and by the multiplication rule that is six \(x\)s.

\((x^3)^2 = x^3 \times x^3 = x^6\)

Negative Powers

This step combines multiplying and dividing. Multiplying by a negative power subtracts from the power — which is exactly what division does. And dividing is multiplying by a fraction, which is why a negative power is a fraction.

\(x^{-n} = \dfrac{1}{x^n}\)

Log Definition

Start the second half with what a log means: \(\log_a b\) is the power of \(a\) that gives \(b\). Both identities below say exactly that, from each direction.

\(a^{\log_a b} = b\)

\(\log_a(a^b) = b\)

Adding Logs

The mirror image of multiplying exponentials: there, multiplying the values added the powers — here, adding the logs multiplies the values.

\(\log_a b + \log_a c = \log_a(bc)\)

Subtracting Logs

The same connection, this time to dividing exponentials: subtracting logs divides the values.

\(\log_a b - \log_a c = \log_a\!\left(\dfrac{b}{c}\right)\)

Extracting Powers

The counterpart of powers of powers: a power inside a log comes out the front as a multiplier.

\(\log_a(b^c) = c\,\log_a b\)

Change of Base

This one is a bit of its own thing, and the most complicated of the rules — which is why it comes last, once everything else is comfortable.

\(\log_a b = \dfrac{\log_c b}{\log_c a}\)

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