Precalculus Examples

$\log_{9} (x^{8} \sqrt{\frac{y}{z^{7}}})$

Step 1

Rewrite $\log_{9} (x^{8} \sqrt{\frac{y}{z^{7}}})$ as $\log_{9} (x^{8}) + \log_{9} (\sqrt{\frac{y}{z^{7}}})$ .

$\log_{9} (x^{8}) + \log_{9} (\sqrt{\frac{y}{z^{7}}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{y}{z^{7}}}$ as ${(\frac{y}{z^{7}})}^{\frac{1}{2}}$ .

$\log_{9} (x^{8}) + \log_{9} ({(\frac{y}{z^{7}})}^{\frac{1}{2}})$

Step 3

Expand $\log_{9} (x^{8})$ by moving $8$ outside the logarithm.

$8 \log_{9} (x) + \log_{9} ({(\frac{y}{z^{7}})}^{\frac{1}{2}})$

Step 4

Expand $\log_{9} ({(\frac{y}{z^{7}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$8 \log_{9} (x) + \frac{1}{2} \log_{9} (\frac{y}{z^{7}})$

Step 5

Rewrite $\log_{9} (\frac{y}{z^{7}})$ as $\log_{9} (y) - \log_{9} (z^{7})$ .

$8 \log_{9} (x) + \frac{1}{2} (\log_{9} (y) - \log_{9} (z^{7}))$

Step 6

Expand $\log_{9} (z^{7})$ by moving $7$ outside the logarithm.

$8 \log_{9} (x) + \frac{1}{2} (\log_{9} (y) - (7 \log_{9} (z)))$

Step 7

Multiply $7$ by $- 1$ .

$8 \log_{9} (x) + \frac{1}{2} (\log_{9} (y) - 7 \log_{9} (z))$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$8 \log_{9} (x) + \frac{1}{2} \log_{9} (y) + \frac{1}{2} (- 7 \log_{9} (z))$

Step 8.2

Combine $\frac{1}{2}$ and $\log_{9} (y)$ .

$8 \log_{9} (x) + \frac{\log_{9} (y)}{2} + \frac{1}{2} (- 7 \log_{9} (z))$

Step 8.3

Multiply $\frac{1}{2} (- 7 \log_{9} (z))$ .

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Step 8.3.1

Combine $- 7$ and $\frac{1}{2}$ .

$8 \log_{9} (x) + \frac{\log_{9} (y)}{2} + \frac{- 7}{2} \log_{9} (z)$

Step 8.3.2

Combine $\frac{- 7}{2}$ and $\log_{9} (z)$ .

$8 \log_{9} (x) + \frac{\log_{9} (y)}{2} + \frac{- 7 \log_{9} (z)}{2}$

Step 8.4

Move the negative in front of the fraction.

$8 \log_{9} (x) + \frac{\log_{9} (y)}{2} - \frac{7 \log_{9} (z)}{2}$