# Why we use logarithm?

Mar 30, 2017

Some thoughts...

#### Explanation:

I am not sure what the context of your question is, but logarithms - especially the natural logarithm - occur naturally in a variety of circumstances.

The natural logarithm $\ln x$ is the inverse of the exponential function ${e}^{x}$, which has lots of interesting properties.

When you get onto calculus, you will find that the natural logarithm occurs as the integral of $\frac{1}{x}$...

$\int \frac{1}{x} \mathrm{dx} = \ln \left\mid x \right\mid + C$

Logarithms are the basis upon which slide rules work.

On a practical note, logarithms allow us to express on a linear scale the measure of physical properties that vary exponentially.

For example, the pH of a solution is $- {\log}_{10}$ of the hydrogen ion concentration. In pure water there are about ${10}^{- 7}$ parts $O {H}^{-}$ ions and ${10}^{- 7}$ parts ${H}^{+}$ (actually ${H}_{3} {O}^{+}$) ions. So neutral pH is $7$. As a solution becomes more acidic, the concentration of ${H}^{+}$ increases and the concentration of $O {H}^{-}$ ions decreases in proportion. So an acid with pH $1$ has a ${10}^{- 1}$ concentration of ${H}^{+}$ (i.e. one part in $10$) and a ${10}^{- 13}$ concentration of $O {H}^{-}$.

Another example would be decibels, which are a logarithmic measure of loudness.

If you are trying to create a model of experimental data that you suspect is exponential, then you would typically take the logarithm of measured values against a variety of input values, then use linear regression to find a line of best fit. Then reverse the logarithm by taking the exponential of that line to get an exponential curve of best fit for your data.