Precalculus Examples

$\log_{b} (\sqrt{\frac{x y^{4}}{z^{5}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x y^{4}}{z^{5}}}$ as ${(\frac{x y^{4}}{z^{5}})}^{\frac{1}{2}}$ .

$\log_{b} ({(\frac{x y^{4}}{z^{5}})}^{\frac{1}{2}})$

Step 2

Expand $\log_{b} ({(\frac{x y^{4}}{z^{5}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{b} (\frac{x y^{4}}{z^{5}})$

Step 3

Rewrite $\log_{b} (\frac{x y^{4}}{z^{5}})$ as $\log_{b} (x y^{4}) - \log_{b} (z^{5})$ .

$\frac{1}{2} (\log_{b} (x y^{4}) - \log_{b} (z^{5}))$

Step 4

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

$\frac{1}{2} (\log_{b} (x y^{4}) - (5 \log_{b} (z)))$

Step 5

Multiply $5$ by $- 1$ .

$\frac{1}{2} (\log_{b} (x y^{4}) - 5 \log_{b} (z))$

Step 6

Rewrite $\log_{b} (x y^{4})$ as $\log_{b} (x) + \log_{b} (y^{4})$ .

$\frac{1}{2} (\log_{b} (x) + \log_{b} (y^{4}) - 5 \log_{b} (z))$

Step 7

Expand $\log_{b} (y^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{2} (\log_{b} (x) + 4 \log_{b} (y) - 5 \log_{b} (z))$

Step 8

Apply the distributive property.

$\frac{1}{2} \log_{b} (x) + \frac{1}{2} (4 \log_{b} (y)) + \frac{1}{2} (- 5 \log_{b} (z))$

Step 9

Simplify.

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Step 9.1

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

$\frac{\log_{b} (x)}{2} + \frac{1}{2} (4 \log_{b} (y)) + \frac{1}{2} (- 5 \log_{b} (z))$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Factor $2$ out of $4 \log_{b} (y)$ .

$\frac{\log_{b} (x)}{2} + \frac{1}{2} (2 (2 \log_{b} (y))) + \frac{1}{2} (- 5 \log_{b} (z))$

Step 9.2.2

Cancel the common factor.

$\frac{\log_{b} (x)}{2} + \frac{1}{2} (2 (2 \log_{b} (y))) + \frac{1}{2} (- 5 \log_{b} (z))$

Step 9.2.3

Rewrite the expression.

$\frac{\log_{b} (x)}{2} + 2 \log_{b} (y) + \frac{1}{2} (- 5 \log_{b} (z))$

Step 9.3

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

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Step 9.3.1

Combine $- 5$ and $\frac{1}{2}$ .

$\frac{\log_{b} (x)}{2} + 2 \log_{b} (y) + \frac{- 5}{2} \log_{b} (z)$

Step 9.3.2

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

$\frac{\log_{b} (x)}{2} + 2 \log_{b} (y) + \frac{- 5 \log_{b} (z)}{2}$

Step 10

Move the negative in front of the fraction.

$\frac{\log_{b} (x)}{2} + 2 \log_{b} (y) - \frac{5 \log_{b} (z)}{2}$