MathematicsChapter 1: Functions and Graphs

Exponential and Logarithmic Graphs

Exponential and logarithmic functions with the same valid base are inverses and reflect in y=x.

Learning objectives

State the rule for exponential and logarithmic graphs.
Solve a exponential and logarithmic graphs problem step by step.
Check the result, restrictions, notation, and units.

Definition

Exponential and logarithmic functions with the same valid base are inverses and reflect in y=x.

Mathematical form

y=x. Worked example

x=8 gives x

Where / symbols

Exponential and logarithmic graphs: Exponential and logarithmic functions with the same valid base are inverses and reflect in y=x.

CONCEPT AND RULE

Exponential and logarithmic functions with the same valid base are inverses and reflect in y=x.

METHOD

For exponential and logarithmic graphs, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.

WORKED EXAMPLE

2^x=8 gives x=3, while log₂(8)=3.

COMMON MISTAKE

Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.

How to solve exponential and logarithmic graphs

Identify the mathematical objects, givens, restrictions, and required result.
Choose and write the relevant definition, theorem, or formula.
Substitute with brackets and carry out one justified step per line.
Simplify, state the final answer clearly, and verify it in the original setting.

Worked example

Worked example:
2^x=8 gives x=3, while log₂(8)=3.

In simple words

Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.

Key points

Exponential and logarithmic graphs
rule
worked example

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.