Precalculus Examples

$2^{\frac{2}{\log_{5} (x)}}$

Step 1

Rewrite $\log_{5} (x)$ using the change of base formula.

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Step 1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$2^{\frac{2}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}}$

Step 1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$2^{\frac{2}{\frac{\log (x)}{\log (5)}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$2^{2 \frac{\log (5)}{\log (x)}}$

Step 3

Multiply $2 \frac{\log (5)}{\log (x)}$ .

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Step 3.1

Combine $2$ and $\frac{\log (5)}{\log (x)}$ .

$2^{\frac{2 \log (5)}{\log (x)}}$

Step 3.2

Simplify $2 \log (5)$ by moving $2$ inside the logarithm.

$2^{\frac{\log (5^{2})}{\log (x)}}$

Step 4

Raise $5$ to the power of $2$ .

$2^{\frac{\log (25)}{\log (x)}}$