Precalculus Examples

$\log_{b} (\sqrt{\frac{4}{125}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{4}{125}}$ as ${(\frac{4}{125})}^{\frac{1}{2}}$ .

$\log_{b} ({(\frac{4}{125})}^{\frac{1}{2}})$

Step 2

Expand $\log_{b} ({(\frac{4}{125})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{b} (\frac{4}{125})$

Step 3

Rewrite $\log_{b} (\frac{4}{125})$ as $\log_{b} (4) - \log_{b} (125)$ .

$\frac{1}{2} (\log_{b} (4) - \log_{b} (125))$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\log_{b} (4)$ as $\log_{b} (2^{2})$ .

$\frac{1}{2} (\log_{b} (2^{2}) - \log_{b} (125))$

Step 4.2

Expand $\log_{b} (2^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{2} (2 \log_{b} (2) - \log_{b} (125))$

Step 4.3

Rewrite $\log_{b} (125)$ as $\log_{b} (5^{3})$ .

$\frac{1}{2} (2 \log_{b} (2) - \log_{b} (5^{3}))$

Step 4.4

Expand $\log_{b} (5^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (2 \log_{b} (2) - (3 \log_{b} (5)))$

Step 4.5

Multiply $3$ by $- 1$ .

$\frac{1}{2} (2 \log_{b} (2) - 3 \log_{b} (5))$

Step 5

Apply the distributive property.

$\frac{1}{2} (2 \log_{b} (2)) + \frac{1}{2} (- 3 \log_{b} (5))$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Factor $2$ out of $2 \log_{b} (2)$ .

$\frac{1}{2} (2 (\log_{b} (2))) + \frac{1}{2} (- 3 \log_{b} (5))$

Step 6.2

Cancel the common factor.

$\frac{1}{2} (2 \log_{b} (2)) + \frac{1}{2} (- 3 \log_{b} (5))$

Step 6.3

Rewrite the expression.

$\log_{b} (2) + \frac{1}{2} (- 3 \log_{b} (5))$

Step 7

Multiply $\frac{1}{2} (- 3 \log_{b} (5))$ .

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Step 7.1

Combine $- 3$ and $\frac{1}{2}$ .

$\log_{b} (2) + \frac{- 3}{2} \log_{b} (5)$

Step 7.2

Combine $\frac{- 3}{2}$ and $\log_{b} (5)$ .

$\log_{b} (2) + \frac{- 3 \log_{b} (5)}{2}$

Step 8

Move the negative in front of the fraction.

$\log_{b} (2) - \frac{3 \log_{b} (5)}{2}$