Precalculus Examples

$\log_{2} (\frac{2 \sqrt{3}}{5})$

Step 1

Rewrite $\log_{2} (\frac{2 \sqrt{3}}{5})$ as $\log_{2} (2 \sqrt{3}) - \log_{2} (5)$ .

$\log_{2} (2 \sqrt{3}) - \log_{2} (5)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\log_{2} (2 \cdot 3^{\frac{1}{2}}) - \log_{2} (5)$

Step 3

Rewrite $\log_{2} (2 \cdot 3^{\frac{1}{2}})$ as $\log_{2} (2) + \log_{2} (3^{\frac{1}{2}})$ .

$\log_{2} (2) + \log_{2} (3^{\frac{1}{2}}) - \log_{2} (5)$

Step 4

Expand $\log_{2} (3^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{2} (2) + \frac{1}{2} \log_{2} (3) - \log_{2} (5)$

Step 5

Logarithm base $2$ of $2$ is $1$ .

$1 + \frac{1}{2} \log_{2} (3) - \log_{2} (5)$

Step 6

Combine $\frac{1}{2}$ and $\log_{2} (3)$ .

$1 + \frac{\log_{2} (3)}{2} - \log_{2} (5)$