One of the most widely used exponential functions would be the one used to find compound interest.
For an amount P deposited at an annual rate of interest r for n years, the amount after n years is given by:
Pn = P0*( 1+ r)^n, the total amount as can be seen grows at an exponential rate. The interest is given by the total amount at any time decreased by the initial amount.
A logarithmic function is used to find the value for the time elapsed if the amount present , the rate of interest and the initial amount are given.
Pn = P0*( 1+ r)^n
=> Pn/P0 = (1+r)^n
=> log( Pn/P0) = log ( 1+ r)^n
=> log( Pn/P0) = n* log ( 1+ r)
=> n = log (Pn/P0)/ ln(1+r)
substituting Pn, P0 and r gives us n.