Answer
$$D = \left( { - \infty ,\infty } \right),{\text{ }}R = \left( {0,\infty } \right)$$
Work Step by Step
$$\eqalign{
& {\text{Let }}f\left( x \right) = \ln x.{\text{ Its inverse is }}{f^{ - 1}}\left( x \right) = {e^x} \cr
& {\text{We know that the range of }}f\left( x \right) = \ln x{\text{ is }}R = \left( { - \infty ,\infty } \right),{\text{ then}} \cr
& {\text{the domain of its inverse }}{f^{ - 1}}\left( x \right) = {e^x}{\text{ is }}D = \left( { - \infty ,\infty } \right) \cr
& \cr
& {\text{We know that the domain of }}f\left( x \right) = \ln x{\text{ is }}D = \left( {0,\infty } \right),{\text{ then}} \cr
& {\text{the range of its inverse }}{f^{ - 1}}\left( x \right) = {e^x}{\text{ is }}R = \left( {0,\infty } \right) \cr} $$
