Precalculus Examples

$\log_{m} (\frac{\sqrt{2 r^{7}}}{z^{3}})$

Step 1

Rewrite $\log_{m} (\frac{\sqrt{2 r^{7}}}{z^{3}})$ as $\log_{m} (\sqrt{2 r^{7}}) - \log_{m} (z^{3})$ .

$\log_{m} (\sqrt{2 r^{7}}) - \log_{m} (z^{3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 r^{7}}$ as ${(2 r^{7})}^{\frac{1}{2}}$ .

$\log_{m} ({(2 r^{7})}^{\frac{1}{2}}) - \log_{m} (z^{3})$

Step 3

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

$\frac{1}{2} \log_{m} (2 r^{7}) - \log_{m} (z^{3})$

Step 4

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

$\frac{1}{2} \log_{m} (2 r^{7}) - (3 \log_{m} (z))$

Step 5

Multiply $3$ by $- 1$ .

$\frac{1}{2} \log_{m} (2 r^{7}) - 3 \log_{m} (z)$

Step 6

Rewrite $\log_{m} (2 r^{7})$ as $\log_{m} (2) + \log_{m} (r^{7})$ .

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

Step 7

Expand $\log_{m} (r^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{2} (\log_{m} (2) + 7 \log_{m} (r)) - 3 \log_{m} (z)$

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

Apply the distributive property.

$\frac{1}{2} \log_{m} (2) + \frac{1}{2} (7 \log_{m} (r)) - 3 \log_{m} (z)$

Step 8.2

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

$\frac{\log_{m} (2)}{2} + \frac{1}{2} (7 \log_{m} (r)) - 3 \log_{m} (z)$

Step 8.3

Multiply $\frac{1}{2} (7 \log_{m} (r))$ .

Tap for more steps...

Step 8.3.1

Combine $7$ and $\frac{1}{2}$ .

$\frac{\log_{m} (2)}{2} + \frac{7}{2} \log_{m} (r) - 3 \log_{m} (z)$

Step 8.3.2

Combine $\frac{7}{2}$ and $\log_{m} (r)$ .

$\frac{\log_{m} (2)}{2} + \frac{7 \log_{m} (r)}{2} - 3 \log_{m} (z)$