Why we use logarithm?

1 Answer
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 #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.