Precalculus Examples

$\log_{9} (x^{6} \sqrt{\frac{y}{z^{5}}})$

Step 1

Rewrite $\log_{9} (x^{6} \sqrt{\frac{y}{z^{5}}})$ as $\log_{9} (x^{6}) + \log_{9} (\sqrt{\frac{y}{z^{5}}})$ .

$\log_{9} (x^{6}) + \log_{9} (\sqrt{\frac{y}{z^{5}}})$

Step 2

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

$\log_{9} (x^{6}) + \log_{9} ({(\frac{y}{z^{5}})}^{\frac{1}{2}})$

Step 3

Expand $\log_{9} (x^{6})$ by moving $6$ outside the logarithm.

$6 \log_{9} (x) + \log_{9} ({(\frac{y}{z^{5}})}^{\frac{1}{2}})$

Step 4

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

$6 \log_{9} (x) + \frac{1}{2} \log_{9} (\frac{y}{z^{5}})$

Step 5

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

$6 \log_{9} (x) + \frac{1}{2} (\log_{9} (y) - \log_{9} (z^{5}))$

Step 6

Expand $\log_{9} (z^{5})$ by moving $5$ outside the logarithm.

$6 \log_{9} (x) + \frac{1}{2} (\log_{9} (y) - (5 \log_{9} (z)))$

Step 7

Multiply $5$ by $- 1$ .

$6 \log_{9} (x) + \frac{1}{2} (\log_{9} (y) - 5 \log_{9} (z))$

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

Apply the distributive property.

$6 \log_{9} (x) + \frac{1}{2} \log_{9} (y) + \frac{1}{2} (- 5 \log_{9} (z))$

Step 8.2

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

$6 \log_{9} (x) + \frac{\log_{9} (y)}{2} + \frac{1}{2} (- 5 \log_{9} (z))$

Step 8.3

Multiply $\frac{1}{2} (- 5 \log_{9} (z))$ .

Tap for more steps...

Step 8.3.1

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

$6 \log_{9} (x) + \frac{\log_{9} (y)}{2} + \frac{- 5}{2} \log_{9} (z)$

Step 8.3.2

Combine $\frac{- 5}{2}$ and $\log_{9} (z)$ .

$6 \log_{9} (x) + \frac{\log_{9} (y)}{2} + \frac{- 5 \log_{9} (z)}{2}$

Step 8.4

Move the negative in front of the fraction.

$6 \log_{9} (x) + \frac{\log_{9} (y)}{2} - \frac{5 \log_{9} (z)}{2}$