Question

If `5^x = 3`, what does `5^-(2x)` equal?

Answer 1

`1/9`

Explanation:

1. First, take the log (base 10) of both sides.

`5^x = 3`
`log(5^x) = log(3)`

2. When you have an exponent inside of a log, you can take that exponent out and multiply it by the rest of the log.

`log(x^y) = y * log(x)`

This is known as the power rule of logarithms.

`log(5^x) = log(3)`
`x* log(5) = log(3)`

3. Now divide both sides by `log(5)` to find `x`.

`(x* cancel(log(5)))/cancel(log(5)) = log(3)/log(5)`

`x = log(3)/log(5)`

4. Now, plug `x` (in terms of the logs) into the second expression.

`5^(-(2* log(3)/log(5))`

Let's evaluate the power first. (I used a calculator here.)

`-(2*log(3)/log(5)) ~~ -1.365212389`

`5^-1.365212389 = bar (.1) = 1/9`

So `5^(-2x) = 1/9`.

Hope this helps! :)