Precalculus Examples

$\log_{5} (\sqrt{3125})$

Step 1

Rewrite $3125$ as $25^{2} \cdot 5$ .

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Step 1.1

Factor $625$ out of $3125$ .

$\log_{5} (\sqrt{625 (5)})$

Step 1.2

Rewrite $625$ as $25^{2}$ .

$\log_{5} (\sqrt{25^{2} \cdot 5})$

Step 2

Pull terms out from under the radical.

$\log_{5} (25 \sqrt{5})$

Step 3

Rewrite $\log_{5} (25 \sqrt{5})$ 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 $b = 10$ .

$\frac{\log (25 \sqrt{5})}{\log (5)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (25 \sqrt{5})}{\log (5)}$

Decimal Form:

$2.5$