Precalculus Examples

$\log_{b} (\sqrt{\frac{13}{25}})$

Step 1

Rewrite $\sqrt{\frac{13}{25}}$ as $\frac{\sqrt{13}}{\sqrt{25}}$ .

$\log_{b} (\frac{\sqrt{13}}{\sqrt{25}})$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $25$ as $5^{2}$ .

$\log_{b} (\frac{\sqrt{13}}{\sqrt{5^{2}}})$

Step 2.2

Pull terms out from under the radical, assuming positive real numbers.

$\log_{b} (\frac{\sqrt{13}}{5})$

Step 3

Rewrite $\log_{b} (\frac{\sqrt{13}}{5})$ using the change of base formula.

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Step 3.1

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 $0$ .

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

Step 3.2

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

$\frac{\log (\frac{\sqrt{13}}{5})}{\log (b)}$