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

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Answer 1 · 2017-06-28T14:53:36

$\frac{1}{9}$ $1/9$

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

$5^{x} = 3$ $5^x = 3$
$log (5^{x}) = log (3)$ $log(5^x) = log(3)$

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.

$log (x^{y}) = y \cdot log (x)$ $log(x^y) = y * log(x)$

This is known as the power rule of logarithms.

$log (5^{x}) = log (3)$ $log(5^x) = log(3)$
$x \cdot log (5) = log (3)$ $x* log(5) = log(3)$

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

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

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

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! :)

If 5^x = 35x=35^x = 3, what does 5^-(2x)5−(2x)5^-(2x) equal?