Question

How do you solve `(1/2)^x =16^2`?

Answer 1

`x=-8`

Explanation:

You could use logarithms, but since `1/2` and `16` are both powers of 2, it's easier if you use that fact.

We have `1/2=2^{-1}` and `16=2^{4}`. Therefore the equation `(1/2)^{x}=16^{2}` can be written as `(2^{-1})^{x}=(2^{4})^{2}`, or `2^{-x}=2^{8}`. This implies that `-x=8` so that `x=-8` (after all, `2^{z}` is a one-to-one function of the variable `z` (just to pick a different letter than `x`)).

You can check the answer like this:

`(1/2)^{-8}=2^{8}=256=16^2`