The logarithmic function

Exponential functions and logarithms: Logarithmic functions

The logarithmic function

Earlier, we had a look at exponential functions. We'll now have a look at logarithms. The logarithm can be seen as the inverse, or opposite, of the exponential function. Let's take the function $\blue{a}^x$ ; it might be interesting to solve for $x$ in the equation $\blue{a}^x=\green{b}$ . When solving this equation, we'll use the logarithm.

Logarithm

The number $\log_{\blue{a}}\left(\green{b}\right)$ is the exponent to which the number $\blue{a}$ must be raised to get $\green{b}$ . We call this the logarithm. Therefore

$\begin{array}{lcr}\log_{\blue{a}}\left(\green{b}\right)=x&\text{ gives }&\blue{a}^x=\green{b}\end{array}$

The number $\blue{a}$ is called the base of the logarithm. The number $\blue{a}$ has to be positive and not equal to $1$ , the number $\green{b}$ has to be positive.

Examples

$\begin{array}{lcrl}\log_{\blue{2}}\left(\green{8}\right)&=&3&\text{because }\blue{2}^3=\green{8} \\ \log_{\blue{4}}\left(\green{\frac{1}{16}}\right)&=&-2&\text{because }\blue{4}^{-2}=\green{\frac{1}{16}} \\ \log_{\blue{5}}\left(\green{\sqrt{5}}\right)&=&\frac{1}{2}&\text{because }\blue{5}^{\frac{1}{2}}=\green{\sqrt{5}} \\ \end{array}$

Etymology

The word logarithm is a combination of two Greek words 'logos' and 'arithmos'. 'Arithmos' means number. 'Logos' means ratio. The term was coined by Scottish mathematician John Napier in 1614.

We can derive two important rules from the definition of logarithm.

$\begin{array}{rcl}\blue{a}^{\log_{\blue{a}}\left(\green{b}\right)}&=&\green{b}\end{array}$

Example

$\begin{array}{lccr}\blue{3}^{\log_{\blue{3}}\left(\green{9}\right)}&=&\green{9}\end{array}$

$\begin{array}{rcl}\log_{\blue{a}}\left(\blue{a}^{\green{b}}\right)&=&\green{b}\end{array}$

Example

$\begin{array}{lccrr}\log_{\blue{a}}\left(1\right)&=&\log_{\blue{a}}\left(\blue{a}^\green{0}\right)&=&\green{0}\end{array}$

Simplify $\log_{2}\left(16\right).$

$4$

$\begin{array}{rcl} \log_{2}\left(16\right)&=&\log_{2}\left(2^4 \right)\\ &&\quad \blue{\text{identify }x\text{ such that }2^x=16}\\ &=& 4\\ &&\quad \blue{\log_{a}\left(a^b\right)=b} \end{array}$

New example