What f-1(x) of f(x)=(1/3)^x

Answer 1

You want the inverse of:

y = (1/3)^x ?????
hmmm maybe try
x = (1/3)^y
ln x = y ln(1/3)

y = ln x /ln(1/3) = -ln x ln 3

y = - 1.1 ln x approximately

Answer 2

or, to avoid introducing a new base,

f^-1(x) = log_1/3x = -log₃x

log and exponentiation are inverses, just like

+ and -
* and /
square root and square

We just don't have any special symbols to denote the operation.