Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

$\frac{1}{2} \log_{b} (x) - (3 \log_{b} (y))$

Step 5

Multiply $3$ by $- 1$ .

$\frac{1}{2} \log_{b} (x) - 3 \log_{b} (y)$

Step 6

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

$\frac{\log_{b} (x)}{2} - 3 \log_{b} (y)$