Precalculus Examples

$\log_{4} (\frac{7 \sqrt{6}}{5})$

Step 1

Rewrite $\log_{4} (\frac{7 \sqrt{6}}{5})$ as $\log_{4} (7 \sqrt{6}) - \log_{4} (5)$ .

$\log_{4} (7 \sqrt{6}) - \log_{4} (5)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\log_{4} (7 \cdot 6^{\frac{1}{2}}) - \log_{4} (5)$

Step 3

Rewrite $\log_{4} (7 \cdot 6^{\frac{1}{2}})$ as $\log_{4} (7) + \log_{4} (6^{\frac{1}{2}})$ .

$\log_{4} (7) + \log_{4} (6^{\frac{1}{2}}) - \log_{4} (5)$

Step 4

Expand $\log_{4} (6^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{4} (7) + \frac{1}{2} \log_{4} (6) - \log_{4} (5)$

Step 5

Rewrite $\log_{4} (6)$ as $\log_{4} (2 (3))$ .

$\log_{4} (7) + \frac{1}{2} \log_{4} (2 (3)) - \log_{4} (5)$

Step 6

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

$\log_{4} (7) + \frac{1}{2} (\log_{4} (2) + \log_{4} (3)) - \log_{4} (5)$

Step 7

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$\log_{4} (7) + \frac{1}{2} (\frac{1}{2} + \log_{4} (3)) - \log_{4} (5)$

Step 8

Apply the distributive property.

$\log_{4} (7) + \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \log_{4} (3) - \log_{4} (5)$

Step 9

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 9.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\log_{4} (7) + \frac{1}{2 \cdot 2} + \frac{1}{2} \log_{4} (3) - \log_{4} (5)$

Step 9.2

Multiply $2$ by $2$ .

$\log_{4} (7) + \frac{1}{4} + \frac{1}{2} \log_{4} (3) - \log_{4} (5)$

Step 10

Combine $\frac{1}{2}$ and $\log_{4} (3)$ .

$\log_{4} (7) + \frac{1}{4} + \frac{\log_{4} (3)}{2} - \log_{4} (5)$