Precalculus Examples

$\log_{16} (\sqrt[3]{\frac{13}{q^{2} p}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{13}{q^{2} p}}$ as ${(\frac{13}{q^{2} p})}^{\frac{1}{3}}$ .

$\log_{16} ({(\frac{13}{q^{2} p})}^{\frac{1}{3}})$

Step 2

Expand $\log_{16} ({(\frac{13}{q^{2} p})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{16} (\frac{13}{q^{2} p})$

Step 3

Rewrite $\log_{16} (\frac{13}{q^{2} p})$ as $\log_{16} (13) - \log_{16} (q^{2} p)$ .

$\frac{1}{3} (\log_{16} (13) - \log_{16} (q^{2} p))$

Step 4

Rewrite $\log_{16} (q^{2} p)$ as $\log_{16} (q^{2}) + \log_{16} (p)$ .

$\frac{1}{3} (\log_{16} (13) - (\log_{16} (q^{2}) + \log_{16} (p)))$

Step 5

Expand $\log_{16} (q^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{3} (\log_{16} (13) - (2 \log_{16} (q) + \log_{16} (p)))$

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Apply the distributive property.

$\frac{1}{3} (\log_{16} (13) - (2 \log_{16} (q)) - \log_{16} (p))$

Step 6.2

Multiply $2$ by $- 1$ .

$\frac{1}{3} (\log_{16} (13) - 2 \log_{16} (q) - \log_{16} (p))$

Step 7

Apply the distributive property.

$\frac{1}{3} \log_{16} (13) + \frac{1}{3} (- 2 \log_{16} (q)) + \frac{1}{3} (- \log_{16} (p))$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Combine $\frac{1}{3}$ and $\log_{16} (13)$ .

$\frac{\log_{16} (13)}{3} + \frac{1}{3} (- 2 \log_{16} (q)) + \frac{1}{3} (- \log_{16} (p))$

Step 8.2

Multiply $\frac{1}{3} (- 2 \log_{16} (q))$ .

Tap for more steps...

Step 8.2.1

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

$\frac{\log_{16} (13)}{3} + \frac{- 2}{3} \log_{16} (q) + \frac{1}{3} (- \log_{16} (p))$

Step 8.2.2

Combine $\frac{- 2}{3}$ and $\log_{16} (q)$ .

$\frac{\log_{16} (13)}{3} + \frac{- 2 \log_{16} (q)}{3} + \frac{1}{3} (- \log_{16} (p))$

Step 8.3

Combine $\frac{1}{3}$ and $\log_{16} (p)$ .

$\frac{\log_{16} (13)}{3} + \frac{- 2 \log_{16} (q)}{3} - \frac{\log_{16} (p)}{3}$

Step 9

Move the negative in front of the fraction.

$\frac{\log_{16} (13)}{3} - \frac{2 \log_{16} (q)}{3} - \frac{\log_{16} (p)}{3}$