# Logarithmic and Exponential Functions

> CIE IGCSE Additional Mathematics · CIE IGCSE Add Maths (0606)
> Source: https://www.owlsprep.com/study/cie-0606-u6-overview/
> Weight: 8-10% of total assessment, across both Paper 1 and Paper 2 (both structured written)

This unit builds on prior knowledge of indices to teach core logarithmic and exponential function properties, algebraic manipulation rules, and assessment-aligned problem-solving skills.

**Prerequisites:** CIE IGCSE Mathematics (0580) Indices and Standard Form

## Learning objectives

- Recall and apply laws of exponents and logarithms to simplify algebraic expressions
- Solve linear and non-linear equations containing exponential and logarithmic terms
- Identify key properties and graph shapes of basic exponential and logarithmic functions
- Apply these functions to solve real-world problems including population growth and decay scenarios

## 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](https://www.owlsprep.com/study/cie-0606-u6-exponentials-logarithms-laws-and-equations/) — Covers logarithm definition, log laws, equation solving, graph properties, and real-world application questions for assessment.

## Common pitfalls

- **Wrong:** Mixing up product and power log laws, e.g. writing $\log(a \times b) = \log a \times \log b$
  - Why it fails: The logarithm of a product equals the sum of individual logarithms, not their product, a common algebraic misapplication.
  - Correct: Apply the product law: $\log_a(xy) = \log_a x + \log_a y$, and use the power law for exponents: $\log_a(x^n) = n\log_a x$.
- **Wrong:** Forgetting logarithms are only defined for positive arguments
  - Why it fails: Taking the log of a negative number or zero yields a non-real result, leading to invalid solutions when solving equations.
  - Correct: Always test all solutions to logarithmic equations to confirm every log term has a positive argument, discarding invalid results.
- **Wrong:** Swapping base and argument when converting between exponential and log form, e.g. $a^x = b \to x = \log_b a$
  - Why it fails: The base of the exponential function remains the base of the logarithm in the rewritten form.
  - Correct: Use the correct conversion rule: if $a^x = b$, then $x = \log_a b$.

## Cheatsheet

| Rule Name | Formula |
| --- | --- |
| Exponential-Logarithm Conversion | $a^x = b \iff x = \log_a b$ |
| Product Law of Logarithms | $\log_a(xy) = \log_a x + \log_a y$ |
| Quotient Law of Logarithms | $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$ |
| Power Law of Logarithms | $\log_a(x^n) = n\log_a x$ |
| Change of Base Rule | $\log_a b = \frac{\log_c b}{\log_c a}$ |

## 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.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/cie-0606-u6-overview/