### Introduction

In this Section we revisit the use of exponents. We consider how the expression ${a}^{x}$ is defined when $a$ is a positive number and $x$ is irrational . Previously we have only considered examples in which $x$ is a rational number. We consider these exponential functions $f\left(x\right)={a}^{x}$ in more depth and in particular consider the special case when the base $a$ is the exponential constant $\text{e}$ where :

$\phantom{\rule{2em}{0ex}}\text{e}=2.7182818\dots$

We then examine the behaviour of ${\text{e}}^{x}$ as $x\to \infty$ , called exponential growth and of ${\text{e}}^{-x}$ as $x\to \infty$ called exponential decay .

#### Prerequisites

• have a good knowledge of indices and their laws
• have knowledge of rational and irrational numbers

#### Learning Outcomes

• approximate ${a}^{x}$ when $x$ is irrational
• describe the behaviour of ${a}^{x}$ : in particular the exponential function ${e}^{x}$
• understand the terms exponential growth and exponential decay