The inverse of a logarithmic function is an exponential function. In other words, if $$f(x)$$ is a logarithmic function, then its inverse is an exponential function $$f^{-1}(x)$$. The logarithmic function and its inverse are symmetrical along the line $$y = x$$. To find the inverse of a logarithmic function, follow these steps:
- Replace the function notation $$f(x)$$ with $$y$$.
- Swap the $$x$$s and $$y$$'s.
- Solve the resulting equation for $$y$$.
- Replace $$y$$ with the inverse notation $$f^{-1}(x)$$.
For example, let's find the inverse of the logarithmic function $$f(x) = \log_2(x)$$.
- Replace $$f(x)$$ with $$y$$: $$y = \log_2(x)$$.
- Swap the $$x$$'s and $$y$$'s: $$x = \log_2(y)$$.
- Solve for $$y$$: $$2^x = y$$.
- Replace $$y$$ with the inverse notation $$f^{-1}(x)$$: $$f^{-1}(x) = 2^x$$.
To check if we have the correct inverse, we can graph both the logarithmic function and its inverse on the same $$xy$$-axis. If the graphs are symmetrical along the line $$y = x$$, then we have the correct inverse.
