I cannot figure this out?

Question

Find a formula for the inverse of the function 4^(6)^(x)

Answer 1

First, let f(x) = 4^(6^x). We want to find f^(-1)(x).

Let f(x) = y, so we have

y = 4^(6^x)

Your first step to solve for the functional inverse is to swap the x and y terms.

x = 4^(6^y)

Next, try and solve for y. Since this is an exponent, you have to change this to logarithmic form. Note that if b^a = c, then in logarithmic form, log[base b](c) = a. Therefore

log[base 4](x) = 6^y

Note that, again, we need to bring the y down since it's an exponent. So you have to convert to logarithmic form once again, and so you get

log [base 6] ( log[base 4](x) ) = y

or

y = log [base 6] ( log[base 4](x) )

You then write your concluding statement by replacing y with f^(-1)(x). You state,

Therefore,

f^(-1)(x) = log [base 6] ( log[base 4](x) )

Answer 2

y=4^6^x
log_4 y = 6^x
log_6 (log_4 y) = x
so if f(x) = 4^6^x, then f^(-1)(x) = log_6 (log_4 x).