Precalculus Examples

$\log_{7} (\frac{7 \sqrt{3}}{11 v})$

Step 1

Rewrite $\log_{7} (\frac{7 \sqrt{3}}{11 v})$ as $\log_{7} (7 \sqrt{3}) - \log_{7} (11 v)$ .

$\log_{7} (7 \sqrt{3}) - \log_{7} (11 v)$

Step 2

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

$\log_{7} (7 \cdot 3^{\frac{1}{2}}) - \log_{7} (11 v)$

Step 3

Rewrite $\log_{7} (7 \cdot 3^{\frac{1}{2}})$ as $\log_{7} (7) + \log_{7} (3^{\frac{1}{2}})$ .

$\log_{7} (7) + \log_{7} (3^{\frac{1}{2}}) - \log_{7} (11 v)$

Step 4

Expand $\log_{7} (3^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{7} (7) + \frac{1}{2} \log_{7} (3) - \log_{7} (11 v)$

Step 5

Logarithm base $7$ of $7$ is $1$ .

$1 + \frac{1}{2} \log_{7} (3) - \log_{7} (11 v)$

Step 6

Combine $\frac{1}{2}$ and $\log_{7} (3)$ .

$1 + \frac{\log_{7} (3)}{2} - \log_{7} (11 v)$

Step 7

Rewrite $\log_{7} (11 v)$ as $\log_{7} (11) + \log_{7} (v)$ .

$1 + \frac{\log_{7} (3)}{2} - (\log_{7} (11) + \log_{7} (v))$

Step 8

Apply the distributive property.

$1 + \frac{\log_{7} (3)}{2} - \log_{7} (11) - \log_{7} (v)$