The Exponential Function

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 $f (x) = a^{x}$ in more depth and in particular consider the special case when the base $a$ is the exponential constant $e$ where :

$e = 2.7182818 \dots$

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

1 Exponents revisited

1.1

a^{x}

when

x

is any real number

2 Exponential functions

3 Exponential growth

4 Exponential decay