True or False: if f(x) = 3^x, then f^-1(x) = ln(x)/ln(3) for all x >0

Question

Please Justify why or why not 

Answer 1

let y=f(x)

 

y = 3^x

 

to calculate inverse, we switch x and y and solve for y, where the new y=f-1(x).

x= 3^y   [Note that 3^y is always positive, so x>0 should be assumed. ]

 

Take ln of both sides

ln x = ln (3^y) , assuming x>0

ln x = y * (ln 3)  [property of logarithm]

Finally, solving for y = f^-1 (x) =  ln(x)/ln(3) for all x>0.