Precalculus Examples

$\log_{b} (\sqrt{\frac{23}{79}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{23}{79}}$ as ${(\frac{23}{79})}^{\frac{1}{2}}$ .

$\log_{b} ({(\frac{23}{79})}^{\frac{1}{2}})$

Step 2

Expand $\log_{b} ({(\frac{23}{79})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{b} (\frac{23}{79})$

Step 3

Rewrite $\log_{b} (\frac{23}{79})$ as $\log_{b} (23) - \log_{b} (79)$ .

$\frac{1}{2} (\log_{b} (23) - \log_{b} (79))$

Step 4

Apply the distributive property.

$\frac{1}{2} \log_{b} (23) + \frac{1}{2} (- \log_{b} (79))$

Step 5

Combine $\frac{1}{2}$ and $\log_{b} (23)$ .

$\frac{\log_{b} (23)}{2} + \frac{1}{2} (- \log_{b} (79))$

Step 6

Combine $\frac{1}{2}$ and $\log_{b} (79)$ .

$\frac{\log_{b} (23)}{2} - \frac{\log_{b} (79)}{2}$