Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

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

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