Question

Why we use logarithm?

Answer 1

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 `1/x`...

`int 1/x dx = ln abs(x) + 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 `OH^-` ions and `10^(-7)` parts `H^+` (actually `H_3O^+`) ions. So neutral pH is `7`. As a solution becomes more acidic, the concentration of `H^+` increases and the concentration of `OH^-` 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 `OH^-`.

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.