The Laws of Logarithms (Pure Mathematics 3 – 9709) — Mathematics

Introduction

Logarithms are the inverse operation of exponentiation — they answer the question "to what power must a base be raised to produce a given number?" This relationship sits at the heart of syllabus objective 3.2, which requires you to understand the connection between logarithms and indices and to use the three laws of logarithms fluently.

In 9709 examinations, the laws of logarithms appear in solving exponential equations, simplifying expressions, and proving identities. A firm grasp of this topic is therefore essential for success across the later Pure 3 sections on exponential growth and differential equations.

Core Concept

Logarithms and Indices

The fundamental definition links a logarithm directly to an index (exponent):

a^x = y \iff \log_a y = x \qquad (a > 0,\ a \neq 1,\ y > 0)

Reading this: " $\log_a y$ " asks "what power of $a$ gives $y$ ?" For example, since $2^5 = 32$ , we write $\log_2 32 = 5$ .

Two special bases appear throughout the course:

Common logarithm: base 10, written $\lg x$ (or $\log_{10} x$ ).
Natural logarithm: base $e$ , written $\ln x$ (or $\log_e x$ ).

Important Consequences of the Definition

Because $a^1 = a$ and $a^0 = 1$ :

\log_a a = 1 \qquad \text{and} \qquad \log_a 1 = 0

Because logarithm and exponentiation are inverse operations:

\log_a(a^x) = x \qquad \text{and} \qquad a^{\log_a x} = x

These follow directly from the definition and are used constantly in simplification.

Unlock the full Logarithmic and Exponential Functions note with Nova

You're reading the preview. Unlock the complete note — every worked example, examiner pitfall and practice question — plus 24/7 AI tutoring from Nova that teaches directly from these notes.

Unlock with Nova Explore CAIE A-Level tutoring →

Introduction

Core Concept

Logarithms and Indices

Important Consequences of the Definition

Unlock the full Logarithmic and Exponential Functions note with Nova

Keep learning