Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $9$ by $- 1$ .

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

Step 5

Rewrite $\log (z^{5} y^{\frac{1}{2}})$ as $\log (z^{5}) + \log (y^{\frac{1}{2}})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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