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 :

$\text{e}=2.7182818\dots$

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