Precalculus Examples

$\frac{\log_{b} (\sqrt[3]{x y^{4}})}{z^{5}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x y^{4}}$ as ${(x y^{4})}^{\frac{1}{3}}$ .

$\frac{\log_{b} ({(x y^{4})}^{\frac{1}{3}})}{z^{5}}$

Step 2

Expand $\log_{b} ({(x y^{4})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{\frac{1}{3} \log_{b} (x y^{4})}{z^{5}}$

Step 3

Rewrite $\log_{b} (x y^{4})$ as $\log_{b} (x) + \log_{b} (y^{4})$ .

$\frac{\frac{1}{3} (\log_{b} (x) + \log_{b} (y^{4}))}{z^{5}}$

Step 4

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

$\frac{\frac{1}{3} (\log_{b} (x) + 4 \log_{b} (y))}{z^{5}}$

Step 5

Factor $\frac{1}{3}$ out of $\frac{\frac{1}{3} (\log_{b} (x) + 4 \log_{b} (y))}{z^{5}}$ .

$\frac{1}{3} \cdot \frac{\log_{b} (x) + 4 \log_{b} (y)}{z^{5}}$

Step 6

Multiply $\frac{1}{3}$ by $\frac{\log_{b} (x) + 4 \log_{b} (y)}{z^{5}}$ .

$\frac{\log_{b} (x) + 4 \log_{b} (y)}{3 z^{5}}$