Precalculus Examples

$\log_{4} (\frac{9 \sqrt{7}}{5})$

Step 1

Rewrite $\log_{4} (\frac{9 \sqrt{7}}{5})$ as $\log_{4} (9 \sqrt{7}) - \log_{4} (5)$ .

$\log_{4} (9 \sqrt{7}) - \log_{4} (5)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\log_{4} (9 \cdot 7^{\frac{1}{2}}) - \log_{4} (5)$

Step 3

Rewrite $\log_{4} (9 \cdot 7^{\frac{1}{2}})$ as $\log_{4} (9) + \log_{4} (7^{\frac{1}{2}})$ .

$\log_{4} (9) + \log_{4} (7^{\frac{1}{2}}) - \log_{4} (5)$

Step 4

Expand $\log_{4} (7^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{4} (9) + \frac{1}{2} \log_{4} (7) - \log_{4} (5)$

Step 5

Combine $\frac{1}{2}$ and $\log_{4} (7)$ .

$\log_{4} (9) + \frac{\log_{4} (7)}{2} - \log_{4} (5)$

Step 6

Simplify each term.

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Step 6.1

Rewrite $\log_{4} (9)$ as $\log_{4} (3^{2})$ .

$\log_{4} (3^{2}) + \frac{\log_{4} (7)}{2} - \log_{4} (5)$

Step 6.2

Expand $\log_{4} (3^{2})$ by moving $2$ outside the logarithm.

$2 \log_{4} (3) + \frac{\log_{4} (7)}{2} - \log_{4} (5)$