Precalculus Examples

$\log_{6} (\frac{x^{2} \sqrt[3]{y}}{z^{4}})$

Step 1

Rewrite $\log_{6} (\frac{x^{2} \sqrt[3]{y}}{z^{4}})$ as $\log_{6} (x^{2} \sqrt[3]{y}) - \log_{6} (z^{4})$ .

$\log_{6} (x^{2} \sqrt[3]{y}) - \log_{6} (z^{4})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

$\log_{6} (x^{2} y^{\frac{1}{3}}) - \log_{6} (z^{4})$

Step 3

Expand $\log_{6} (z^{4})$ by moving $4$ outside the logarithm.

$\log_{6} (x^{2} y^{\frac{1}{3}}) - (4 \log_{6} (z))$

Step 4

Multiply $4$ by $- 1$ .

$\log_{6} (x^{2} y^{\frac{1}{3}}) - 4 \log_{6} (z)$

Step 5

Rewrite $\log_{6} (x^{2} y^{\frac{1}{3}})$ as $\log_{6} (x^{2}) + \log_{6} (y^{\frac{1}{3}})$ .

$\log_{6} (x^{2}) + \log_{6} (y^{\frac{1}{3}}) - 4 \log_{6} (z)$

Step 6

Expand $\log_{6} (x^{2})$ by moving $2$ outside the logarithm.

$2 \log_{6} (x) + \log_{6} (y^{\frac{1}{3}}) - 4 \log_{6} (z)$

Step 7

Expand $\log_{6} (y^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$2 \log_{6} (x) + \frac{1}{3} \log_{6} (y) - 4 \log_{6} (z)$

Step 8

Combine $\frac{1}{3}$ and $\log_{6} (y)$ .

$2 \log_{6} (x) + \frac{\log_{6} (y)}{3} - 4 \log_{6} (z)$