Logarithms helps us represent large numbers in very compact form. It can also be used to tell us how many of one number we must multiply to get another number.
The exponential function y=a^x is one of the most important functions in mathematics, physics, and engineering. Applications relating to radioactive decay, bacterial growth, population growth, continuous interest all involve exponential functions. We know that logarithmic and exponential functions are inverses. This means one can change a logarithmic function back to an exponential one by simply finding the inverse function.
With regard to different bases, base 10 is used by chemists in their measurements of pH - the acidity of a liquid. The base 2 is very useful in information theory since computers communicate by using signals which can be coded in terms of 0s and 1s..
The base e, where e=2.718281828, is used in calculus and is probably the most important base. It appears in many business and scientific applications. Logarithms with the base e are called natural logarithms.