Question

How do you solve `7^x= 7^(2^(x^-3))`?

Answer 1

This equality can only be solved with approximations, and the result is something like `x = 1.33670973494950...`

Explanation:

Since the bases are the same (7 in both sides), we have that `7^x = 7^{2^{x^{-3}}}` if and only if the exponents are equal, i.e. if and only if

`x=2^{x^-3}`

and as I said before there is no algebraic way to exactly solve this equation (are you sure you wrote it correctly?). If this is actually the problem you had to solve, you can do nothing better than approximating, and the approximated value is in the answer.