Unit Overview

Logarithmic and Exponential Functions

CIE IGCSE Additional Mathematics· 5 min read 📊 8-10% of total assessment, across both Paper 1 and Paper 2 (both structured written)

1. Unit at a glance

You will first connect your existing understanding of index rules to the formal definition of logarithms, before learning the full set of logarithm laws used to rewrite and simplify complex expressions. You will then apply these rules to solve a range of equations, and learn to recognize the key graphical properties of exponential and logarithmic functions.

This unit contains one comprehensive sub-topic covering all required content for logarithmic and exponential functions:

Exponentials, Logarithms, Laws and Equations

Covers logarithm definition, log laws, equation solving, graph properties, and real-world application questions for assessment.

★★★⏱ 20 min

2. Common Pitfalls

Wrong move:

Mixing up product and power log laws, e.g. writing $lo g (a \times b) = lo g a \times lo g b$

Why:

The logarithm of a product equals the sum of individual logarithms, not their product, a common algebraic misapplication.

Correct move:

Apply the product law: $lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$ , and use the power law for exponents: $lo g_{a} (x^{n}) = n lo g_{a} x$ .

Wrong move:

Forgetting logarithms are only defined for positive arguments

Why:

Taking the log of a negative number or zero yields a non-real result, leading to invalid solutions when solving equations.

Correct move:

Always test all solutions to logarithmic equations to confirm every log term has a positive argument, discarding invalid results.

Wrong move:

Swapping base and argument when converting between exponential and log form, e.g. $a^{x} = b \to x = lo g_{b} a$

Why:

The base of the exponential function remains the base of the logarithm in the rewritten form.

Correct move:

Use the correct conversion rule: if $a^{x} = b$ , then $x = lo g_{a} b$ .

3. Quick Reference Cheatsheet

Rule Name	Formula
Exponential-Logarithm Conversion	$a^{x} = b ⟺ x = lo g_{a} b$
Product Law of Logarithms	$lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$
Quotient Law of Logarithms	$lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y$
Power Law of Logarithms	$lo g_{a} (x^{n}) = n lo g_{a} x$
Change of Base Rule	$lo g_{a} b = \frac{l o g _{c} b}{l o g _{c} a}$

What's Next

Begin your work on this unit by working through the comprehensive sub-topic on exponentials and logarithms, where you will find guided explanations, worked assessment-style examples, and targeted practice to master all required skills. Once you have completed this unit, you will move on to coordinate geometry topics starting with straight line graphs.