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! :)