# What Are Logarithmic Functions?

A logarithmic function is a type of function that is always growing, but very slowly. There are many different logarithmic functions, but the ones you are going to learn in this article are the most common.

The two most common logarithms are the common logarithm, $\mathrm{log}\left(x\right)$ or $\mathrm{lg}\left(x\right)$, and the natural logarithm, $\mathrm{ln}\left(x\right)$.

The functions $\mathrm{log}\left(x\right)$ and $\mathrm{ln}\left(x\right)$ work in exactly the same way when it comes to arithmetic, and they follow the same rules. Their graphs are also about the same. The only difference is that $\mathrm{log}\left(x\right)$ has 10 as its base— that is— $1{0}^{\mathrm{log}x}=x$, while $\mathrm{ln}\left(x\right)$ has $e$ as its base, that is, ${e}^{\mathrm{ln}x}=x$. The natural logarithm got its name because it has the “natural” number $e$ as its base.

Theory

### LogarithmicFunction

The logarithms $\mathrm{ln}\left(\dots \right)$ and $\mathrm{log}\left(\dots \right)$ are functions, so when you take the logarithm of a number ($x=a$) you will get a $y$-value. $\begin{array}{lllllll}\hfill f\left(x\right)& =\mathrm{ln}\left(x\right),\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{2em}{0ex}}x>0& \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill \\ \hfill f\left(x\right)& =\mathrm{log}\left(x\right),\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{2em}{0ex}}x>0& \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill \end{array}$

Note! Since the exponential function is always positive, and the logarithmic function is the “opposite” function of the exponential function, it’s not possible to take the logarithm of a number that is negative or zero.

In this figure, you see the graphs of the common logarithm and the natural logarithm. Both intersect the $x$-axis at $x=1$, but $\mathrm{ln}x$ grows a little faster than $\mathrm{log}x$.

Example 1

The magnitude of an earthquake is measured using the Richter scale. This is a logarithmic scale, and the function is given by

 $R\left(E\right)=\frac{\mathrm{log}E+\text{}1.32\text{}}{\text{}1.44\text{}}$

where $E$ is the energy that is released in kWh, and $R$ is called the Richter magnitude (this is the number you read about in the newspaper when they report the magnitude of the earthquake). The Richter magnitude tells you how strong an earthquake is.

You read in the newspaper one day that an earthquake in Japan released $\text{}1.4\text{}\cdot 1{0}^{9}$ kWh of energy. You wonder what this corresponds on the Richter scale.

To find the Richter magnitude, you have to insert $1.4\cdot 1{0}^{9}$ for $E$, and compute the function value:

$\begin{array}{llll}\hfill R\phantom{\rule{-0.17em}{0ex}}\left(1.4\cdot 1{0}^{9}\right)& =\frac{\mathrm{log}\phantom{\rule{-0.17em}{0ex}}\left(1.4\cdot 1{0}^{9}\right)+1.32}{1.44}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx 7.27\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{lll}\hfill R\phantom{\rule{-0.17em}{0ex}}\left(1.4\cdot 1{0}^{9}\right)=\frac{\mathrm{log}\phantom{\rule{-0.17em}{0ex}}\left(1.4\cdot 1{0}^{9}\right)+1.32}{1.44}\approx 7.27& \phantom{\rule{2em}{0ex}}& \hfill \end{array}$

Thus, the magnitude of the earthquake was equal to $7.27$ on the Richter scale, which is a pretty strong earthquake!