Precalculus Examples

$\log (\sqrt{\frac{y^{9} z}{x^{5}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{y^{9} z}{x^{5}}}$ as ${(\frac{y^{9} z}{x^{5}})}^{\frac{1}{2}}$ .

$\log ({(\frac{y^{9} z}{x^{5}})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{y^{9} z}{x^{5}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{y^{9} z}{x^{5}})$

Step 3

Rewrite $\log (\frac{y^{9} z}{x^{5}})$ as $\log (y^{9} z) - \log (x^{5})$ .

$\frac{1}{2} (\log (y^{9} z) - \log (x^{5}))$

Step 4

Expand $\log (x^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} (\log (y^{9} z) - (5 \log (x)))$

Step 5

Multiply $5$ by $- 1$ .

$\frac{1}{2} (\log (y^{9} z) - 5 \log (x))$

Step 6

Rewrite $\log (y^{9} z)$ as $\log (y^{9}) + \log (z)$ .

$\frac{1}{2} (\log (y^{9}) + \log (z) - 5 \log (x))$

Step 7

Expand $\log (y^{9})$ by moving $9$ outside the logarithm.

$\frac{1}{2} (9 \log (y) + \log (z) - 5 \log (x))$

Step 8

Apply the distributive property.

$\frac{1}{2} (9 \log (y)) + \frac{1}{2} \log (z) + \frac{1}{2} (- 5 \log (x))$

Step 9

Simplify.

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Step 9.1

Multiply $\frac{1}{2} (9 \log (y))$ .

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Step 9.1.1

Combine $9$ and $\frac{1}{2}$ .

$\frac{9}{2} \log (y) + \frac{1}{2} \log (z) + \frac{1}{2} (- 5 \log (x))$

Step 9.1.2

Combine $\frac{9}{2}$ and $\log (y)$ .

$\frac{9 \log (y)}{2} + \frac{1}{2} \log (z) + \frac{1}{2} (- 5 \log (x))$

Step 9.2

Combine $\frac{1}{2}$ and $\log (z)$ .

$\frac{9 \log (y)}{2} + \frac{\log (z)}{2} + \frac{1}{2} (- 5 \log (x))$

Step 9.3

Multiply $\frac{1}{2} (- 5 \log (x))$ .

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Step 9.3.1

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

$\frac{9 \log (y)}{2} + \frac{\log (z)}{2} + \frac{- 5}{2} \log (x)$

Step 9.3.2

Combine $\frac{- 5}{2}$ and $\log (x)$ .

$\frac{9 \log (y)}{2} + \frac{\log (z)}{2} + \frac{- 5 \log (x)}{2}$

Step 10

Move the negative in front of the fraction.

$\frac{9 \log (y)}{2} + \frac{\log (z)}{2} - \frac{5 \log (x)}{2}$