Precalculus Examples

$\log_{4} (\frac{8 \sqrt{5}}{3})$

Step 1

Rewrite $\log_{4} (\frac{8 \sqrt{5}}{3})$ as $\log_{4} (8 \sqrt{5}) - \log_{4} (3)$ .

$\log_{4} (8 \sqrt{5}) - \log_{4} (3)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\log_{4} (8 \cdot 5^{\frac{1}{2}}) - \log_{4} (3)$

Step 3

Rewrite $\log_{4} (8 \cdot 5^{\frac{1}{2}})$ as $\log_{4} (8) + \log_{4} (5^{\frac{1}{2}})$ .

$\log_{4} (8) + \log_{4} (5^{\frac{1}{2}}) - \log_{4} (3)$

Step 4

Expand $\log_{4} (5^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{4} (8) + \frac{1}{2} \log_{4} (5) - \log_{4} (3)$

Step 5

Logarithm base $4$ of $8$ is $\frac{3}{2}$ .

$\frac{3}{2} + \frac{1}{2} \log_{4} (5) - \log_{4} (3)$

Step 6

Combine $\frac{1}{2}$ and $\log_{4} (5)$ .

$\frac{3}{2} + \frac{\log_{4} (5)}{2} - \log_{4} (3)$