Why we use logarithm?
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
When you get onto calculus, you will find that the natural logarithm occurs as the integral of
#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
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.