1 The logarithmic function

In Section 6.3 we introduced the operation of taking logarithms which reverses the operation of exponentiation.

$\text{If}a>0\text{and}a\ne 1\text{then}x=a^y\text{implies}y={log}_{a}x$

In this Section we consider the log function in more detail. We shall concentrate only on the functions $logx$ (i.e. to base 10) and $lnx$ (i.e. to base $\text{e}$ ). The functions $y=logx$ and $y=lnx$ have similar characteristics. We can never choose $x$ as a negative number since $10^y$ and ${\text{e}}^y$ are each always positive. The graphs of $y=logx$ and $y=lnx$ are shown in Figure 5.

Figure 5 :

From the graphs we see that both functions are one-to-one so each has an inverse function - the inverse function of ${log}_{a}x$ is $a^x$ . Let us do this for logs to base 10.