Precalculus Examples

$\log_{5} (\sqrt{125}) - \log_{2} (\sqrt{2})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 1.1.1

Factor $25$ out of $125$ .

$\log_{5} (\sqrt{25 (5)}) - \log_{2} (\sqrt{2})$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

$\log_{5} (5 \sqrt{5}) - \log_{2} (\sqrt{2})$

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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 $0$ .

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

Step 2.1.2

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

$\frac{\log (5 \sqrt{5})}{\log (5)} - \log_{2} (\sqrt{2})$

Step 2.2

Rewrite $\log_{2} (\sqrt{2})$ using the change of base formula.

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Step 2.2.1

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 $0$ .

$\frac{\log (5 \sqrt{5})}{\log (5)} - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2.2

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

$\frac{\log (5 \sqrt{5})}{\log (5)} - \frac{\log (\sqrt{2})}{\log (2)}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (5 \sqrt{5})}{\log (5)} - \frac{\log (\sqrt{2})}{\log (2)}$

Decimal Form:

$1$