How to find the inverse of logarithms?

enter image source here

1 Answer
Mar 1, 2018

#f^(-1)(x)=-ln(x-3)#

Explanation:

Pick up a few values of #x> -3# (as values less than #-3# are not in domain) and corresponding values of #f(x)#, join them to get the graph of #f(x)#. The graph of #f(x)=e^(-x)+3# appears as follows:

graph{e^(-x)+3 [-17.16, 22.84, -1.88, 18.12]}

To find #f^(-1)(x)#, find the value of #x# in terms of #y#

As #y=e^(-x)+3#, we have #e^(-x)=y-3#

and taking natural log on either side

#-x=ln(y-3)# or #x=-ln(y-3)#

Hence #f^(-1)(x)=-ln(x-3)# and if we draw its graph, it appears as

graph{-ln(x-3) [-8.5, 31.5, -3.88, 16.12]}

Note #-# Observe that #f^(-1)(x)# is reflection of #f(x)# in the line #y=x#.