Precalculus Examples

$\log_{a} (\frac{\sqrt{x} y^{3}}{z^{3}})$

Step 1

Rewrite $\log_{a} (\frac{\sqrt{x} y^{3}}{z^{3}})$ as $\log_{a} (\sqrt{x} y^{3}) - \log_{a} (z^{3})$ .

$\log_{a} (\sqrt{x} y^{3}) - \log_{a} (z^{3})$

Step 2

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

$\log_{a} (x^{\frac{1}{2}} y^{3}) - \log_{a} (z^{3})$

Step 3

Expand $\log_{a} (z^{3})$ by moving $3$ outside the logarithm.

$\log_{a} (x^{\frac{1}{2}} y^{3}) - (3 \log_{a} (z))$

Step 4

Multiply $3$ by $- 1$ .

$\log_{a} (x^{\frac{1}{2}} y^{3}) - 3 \log_{a} (z)$

Step 5

Rewrite $\log_{a} (x^{\frac{1}{2}} y^{3})$ as $\log_{a} (x^{\frac{1}{2}}) + \log_{a} (y^{3})$ .

$\log_{a} (x^{\frac{1}{2}}) + \log_{a} (y^{3}) - 3 \log_{a} (z)$

Step 6

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

$\frac{1}{2} \log_{a} (x) + \log_{a} (y^{3}) - 3 \log_{a} (z)$

Step 7

Expand $\log_{a} (y^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} \log_{a} (x) + 3 \log_{a} (y) - 3 \log_{a} (z)$

Step 8

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

$\frac{\log_{a} (x)}{2} + 3 \log_{a} (y) - 3 \log_{a} (z)$