Precalculus Examples

$\frac{\log_{b} (\sqrt{x y^{6}})}{z^{6}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x y^{6}}$ as ${(x y^{6})}^{\frac{1}{2}}$ .

$\frac{\log_{b} ({(x y^{6})}^{\frac{1}{2}})}{z^{6}}$

Step 2

Expand $\log_{b} ({(x y^{6})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{\frac{1}{2} \log_{b} (x y^{6})}{z^{6}}$

Step 3

Rewrite $\log_{b} (x y^{6})$ as $\log_{b} (x) + \log_{b} (y^{6})$ .

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

Step 4

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

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

Step 5

Factor $\frac{1}{2}$ out of $\frac{\frac{1}{2} (\log_{b} (x) + 6 \log_{b} (y))}{z^{6}}$ .

$\frac{1}{2} \cdot \frac{\log_{b} (x) + 6 \log_{b} (y)}{z^{6}}$

Step 6

Multiply $\frac{1}{2}$ by $\frac{\log_{b} (x) + 6 \log_{b} (y)}{z^{6}}$ .

$\frac{\log_{b} (x) + 6 \log_{b} (y)}{2 z^{6}}$