First, replace f(x) by y. This is done to make the rest of the process easier.
[tex]y=6^x-8[/tex]Now, replace every x with y and a every y with an x:
[tex]x=6^y-8[/tex]Now, solve this equation for y. Then, we must move -8 to the left hand side as +8. It yields
[tex]x+8=6^y[/tex]Now, we can apply logarithms in both sides:
[tex]\log (x+8)=\log 6^y[/tex]For the properties of logarithms, we have
[tex]\log 6^y=y\log 6[/tex]then, we have
[tex]\log (x+8)=y\log 6[/tex]and we obtain
[tex]y=\frac{\log(x+8)}{\log6}[/tex]Finally, replace y with f^1. Then, the inverse funcion is
[tex]f^{-1}(x)=\frac{\log(x+8)}{\log6}[/tex]The graphs of the function and its inverse are:
