Precalculus Examples

$\log_{11} (\sqrt{\frac{p q}{8}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{p q}{8}}$ as ${(\frac{p q}{8})}^{\frac{1}{2}}$ .

$\log_{11} ({(\frac{p q}{8})}^{\frac{1}{2}})$

Step 2

Expand $\log_{11} ({(\frac{p q}{8})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{11} (\frac{p q}{8})$

Step 3

Rewrite $\log_{11} (\frac{p q}{8})$ as $\log_{11} (p q) - \log_{11} (8)$ .

$\frac{1}{2} (\log_{11} (p q) - \log_{11} (8))$

Step 4

Rewrite $\log_{11} (8)$ as $\log_{11} (2^{3})$ .

$\frac{1}{2} (\log_{11} (p q) - \log_{11} (2^{3}))$

Step 5

Expand $\log_{11} (2^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (\log_{11} (p q) - (3 \log_{11} (2)))$

Step 6

Multiply $3$ by $- 1$ .

$\frac{1}{2} (\log_{11} (p q) - 3 \log_{11} (2))$

Step 7

Rewrite $\log_{11} (p q)$ as $\log_{11} (p) + \log_{11} (q)$ .

$\frac{1}{2} (\log_{11} (p) + \log_{11} (q) - 3 \log_{11} (2))$

Step 8

Apply the distributive property.

$\frac{1}{2} \log_{11} (p) + \frac{1}{2} \log_{11} (q) + \frac{1}{2} (- 3 \log_{11} (2))$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Combine $\frac{1}{2}$ and $\log_{11} (p)$ .

$\frac{\log_{11} (p)}{2} + \frac{1}{2} \log_{11} (q) + \frac{1}{2} (- 3 \log_{11} (2))$

Step 9.2

Combine $\frac{1}{2}$ and $\log_{11} (q)$ .

$\frac{\log_{11} (p)}{2} + \frac{\log_{11} (q)}{2} + \frac{1}{2} (- 3 \log_{11} (2))$

Step 9.3

Multiply $\frac{1}{2} (- 3 \log_{11} (2))$ .

Tap for more steps...

Step 9.3.1

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{\log_{11} (p)}{2} + \frac{\log_{11} (q)}{2} + \frac{- 3}{2} \log_{11} (2)$

Step 9.3.2

Combine $\frac{- 3}{2}$ and $\log_{11} (2)$ .

$\frac{\log_{11} (p)}{2} + \frac{\log_{11} (q)}{2} + \frac{- 3 \log_{11} (2)}{2}$

Step 10

Move the negative in front of the fraction.

$\frac{\log_{11} (p)}{2} + \frac{\log_{11} (q)}{2} - \frac{3 \log_{11} (2)}{2}$