Precalculus Examples

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

Step 1

Rewrite $\log (z^{5} \sqrt{\frac{{(x + 2)}^{9}}{y}})$ as $\log (z^{5}) + \log (\sqrt{\frac{{(x + 2)}^{9}}{y}})$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Rewrite $\log (\frac{{(x + 2)}^{9}}{y})$ as $\log ({(x + 2)}^{9}) - \log (y)$ .

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

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

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

Step 7.4

Combine the numerators over the common denominator.

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