Precalculus Examples

$\log_{b} (\frac{a \sqrt{b}}{c})$

Step 1

Rewrite $\log_{b} (\frac{a \sqrt{b}}{c})$ as $\log_{b} (a \sqrt{b}) - \log_{b} (c)$ .

$\log_{b} (a \sqrt{b}) - \log_{b} (c)$

Step 2

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

$\log_{b} (a b^{\frac{1}{2}}) - \log_{b} (c)$

Step 3

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

$\log_{b} (a) + \log_{b} (b^{\frac{1}{2}}) - \log_{b} (c)$

Step 4

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

$\log_{b} (a) + \frac{1}{2} \log_{b} (b) - \log_{b} (c)$

Step 5

Logarithm base $b$ of $b$ is $1$ .

$\log_{b} (a) + \frac{1}{2} \cdot 1 - \log_{b} (c)$

Step 6

Multiply $\frac{1}{2}$ by $1$ .

$\log_{b} (a) + \frac{1}{2} - \log_{b} (c)$