Precalculus Examples

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

Step 1

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

$\log (\sqrt{2} y^{5}) - \log (z^{9})$

Step 2

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

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

Step 3

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

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

Step 4

Multiply $9$ by $- 1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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