Precalculus Examples

$\log_{17} (\frac{\sqrt[3]{14}}{n^{2} m})$

Step 1

Rewrite $\log_{17} (\frac{\sqrt[3]{14}}{n^{2} m})$ as $\log_{17} (\sqrt[3]{14}) - \log_{17} (n^{2} m)$ .

$\log_{17} (\sqrt[3]{14}) - \log_{17} (n^{2} m)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{14}$ as $14^{\frac{1}{3}}$ .

$\log_{17} (14^{\frac{1}{3}}) - \log_{17} (n^{2} m)$

Step 3

Expand $\log_{17} (14^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{17} (14) - \log_{17} (n^{2} m)$

Step 4

Rewrite $\log_{17} (14)$ as $\log_{17} (2 (7))$ .

$\frac{1}{3} \log_{17} (2 (7)) - \log_{17} (n^{2} m)$

Step 5

Rewrite $\log_{17} (2 (7))$ as $\log_{17} (2) + \log_{17} (7)$ .

$\frac{1}{3} (\log_{17} (2) + \log_{17} (7)) - \log_{17} (n^{2} m)$

Step 6

Apply the distributive property.

$\frac{1}{3} \log_{17} (2) + \frac{1}{3} \log_{17} (7) - \log_{17} (n^{2} m)$

Step 7

Combine $\frac{1}{3}$ and $\log_{17} (2)$ .

$\frac{\log_{17} (2)}{3} + \frac{1}{3} \log_{17} (7) - \log_{17} (n^{2} m)$

Step 8

Combine $\frac{1}{3}$ and $\log_{17} (7)$ .

$\frac{\log_{17} (2)}{3} + \frac{\log_{17} (7)}{3} - \log_{17} (n^{2} m)$

Step 9

Rewrite $\log_{17} (n^{2} m)$ as $\log_{17} (n^{2}) + \log_{17} (m)$ .

$\frac{\log_{17} (2)}{3} + \frac{\log_{17} (7)}{3} - (\log_{17} (n^{2}) + \log_{17} (m))$

Step 10

Expand $\log_{17} (n^{2})$ by moving $2$ outside the logarithm.

$\frac{\log_{17} (2)}{3} + \frac{\log_{17} (7)}{3} - (2 \log_{17} (n) + \log_{17} (m))$

Step 11

Simplify each term.

Tap for more steps...

Step 11.1

Apply the distributive property.

$\frac{\log_{17} (2)}{3} + \frac{\log_{17} (7)}{3} - (2 \log_{17} (n)) - \log_{17} (m)$

Step 11.2

Multiply $2$ by $- 1$ .

$\frac{\log_{17} (2)}{3} + \frac{\log_{17} (7)}{3} - 2 \log_{17} (n) - \log_{17} (m)$