Precalculus Examples

$\log_{b} (\sqrt{\frac{2}{27}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{2}{27}}$ as ${(\frac{2}{27})}^{\frac{1}{2}}$ .

$\log_{b} ({(\frac{2}{27})}^{\frac{1}{2}})$

Step 2

Expand $\log_{b} ({(\frac{2}{27})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{b} (\frac{2}{27})$

Step 3

Rewrite $\log_{b} (\frac{2}{27})$ as $\log_{b} (2) - \log_{b} (27)$ .

$\frac{1}{2} (\log_{b} (2) - \log_{b} (27))$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\log_{b} (27)$ as $\log_{b} (3^{3})$ .

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

Step 4.2

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

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

Step 4.3

Multiply $3$ by $- 1$ .

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

Step 5

Apply the distributive property.

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

Step 6

Combine $\frac{1}{2}$ and $\log_{b} (2)$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Move the negative in front of the fraction.

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