Find the inverse function of:
[tex]f(x)=\log_3(x+17)[/tex]First, call the function y = f(x):
[tex]y=\log_3(x+17)[/tex]We have to solve for x. We need to apply the definition of logarithms:
If:
[tex]\begin{gathered} a=b^x \\ \text{ Then:} \\ \log_ba=x \end{gathered}[/tex]It can be applied in inverse order, that is, if:
[tex]\begin{gathered} x=\log_ba \\ \\ \text{ Then:} \\ \\ a=b^x \end{gathered}[/tex]Applying the definition above:
[tex]x+17=3^y[/tex]Subtract 17:
[tex]x=3^y-17[/tex]Now we change x for y and vice-versa:
[tex]y=3^x-17[/tex]Finally, substitute y for the inverse of f(x):
[tex]f^{-1}(x)=3^x-17[/tex]