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

1 Answer
Jun 28, 2017

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! :)