Precalculus Examples

$\log_{9} (\frac{2 \sqrt{x}}{y})$

Step 1

Rewrite $\log_{9} (\frac{2 \sqrt{x}}{y})$ as $\log_{9} (2 \sqrt{x}) - \log_{9} (y)$ .

$\log_{9} (2 \sqrt{x}) - \log_{9} (y)$

Step 2

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

$\log_{9} (2 x^{\frac{1}{2}}) - \log_{9} (y)$

Step 3

Rewrite $\log_{9} (2 x^{\frac{1}{2}})$ as $\log_{9} (2) + \log_{9} (x^{\frac{1}{2}})$ .

$\log_{9} (2) + \log_{9} (x^{\frac{1}{2}}) - \log_{9} (y)$

Step 4

Expand $\log_{9} (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{9} (2) + \frac{1}{2} \log_{9} (x) - \log_{9} (y)$

Step 5

Combine $\frac{1}{2}$ and $\log_{9} (x)$ .

$\log_{9} (2) + \frac{\log_{9} (x)}{2} - \log_{9} (y)$