Precalculus Examples

$\log_{6} (\frac{\sqrt{11}}{10})$

Step 1

Rewrite $\log_{6} (\frac{\sqrt{11}}{10})$ as $\log_{6} (\sqrt{11}) - \log_{6} (10)$ .

$\log_{6} (\sqrt{11}) - \log_{6} (10)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\log_{6} (11^{\frac{1}{2}}) - \log_{6} (10)$

Step 3

Expand $\log_{6} (11^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{6} (11) - \log_{6} (10)$

Step 4

Combine $\frac{1}{2}$ and $\log_{6} (11)$ .

$\frac{\log_{6} (11)}{2} - \log_{6} (10)$

Step 5

Rewrite $\log_{6} (10)$ as $\log_{6} (2 (5))$ .

$\frac{\log_{6} (11)}{2} - \log_{6} (2 (5))$

Step 6

Rewrite $\log_{6} (2 (5))$ as $\log_{6} (2) + \log_{6} (5)$ .

$\frac{\log_{6} (11)}{2} - (\log_{6} (2) + \log_{6} (5))$

Step 7

Apply the distributive property.

$\frac{\log_{6} (11)}{2} - \log_{6} (2) - \log_{6} (5)$