Precalculus Examples

$\log_{2} (\sqrt{\frac{3 r^{5}}{z^{3}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{3 r^{5}}{z^{3}}}$ as ${(\frac{3 r^{5}}{z^{3}})}^{\frac{1}{2}}$ .

$\log_{2} ({(\frac{3 r^{5}}{z^{3}})}^{\frac{1}{2}})$

Step 2

Expand $\log_{2} ({(\frac{3 r^{5}}{z^{3}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{2} (\frac{3 r^{5}}{z^{3}})$

Step 3

Rewrite $\log_{2} (\frac{3 r^{5}}{z^{3}})$ as $\log_{2} (3 r^{5}) - \log_{2} (z^{3})$ .

$\frac{1}{2} (\log_{2} (3 r^{5}) - \log_{2} (z^{3}))$

Step 4

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

$\frac{1}{2} (\log_{2} (3 r^{5}) - (3 \log_{2} (z)))$

Step 5

Multiply $3$ by $- 1$ .

$\frac{1}{2} (\log_{2} (3 r^{5}) - 3 \log_{2} (z))$

Step 6

Rewrite $\log_{2} (3 r^{5})$ as $\log_{2} (3) + \log_{2} (r^{5})$ .

$\frac{1}{2} (\log_{2} (3) + \log_{2} (r^{5}) - 3 \log_{2} (z))$

Step 7

Expand $\log_{2} (r^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} (\log_{2} (3) + 5 \log_{2} (r) - 3 \log_{2} (z))$

Step 8

Apply the distributive property.

$\frac{1}{2} \log_{2} (3) + \frac{1}{2} (5 \log_{2} (r)) + \frac{1}{2} (- 3 \log_{2} (z))$

Step 9

Simplify.

Tap for more steps...

Step 9.1

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

$\frac{\log_{2} (3)}{2} + \frac{1}{2} (5 \log_{2} (r)) + \frac{1}{2} (- 3 \log_{2} (z))$

Step 9.2

Multiply $\frac{1}{2} (5 \log_{2} (r))$ .

Tap for more steps...

Step 9.2.1

Combine $5$ and $\frac{1}{2}$ .

$\frac{\log_{2} (3)}{2} + \frac{5}{2} \log_{2} (r) + \frac{1}{2} (- 3 \log_{2} (z))$

Step 9.2.2

Combine $\frac{5}{2}$ and $\log_{2} (r)$ .

$\frac{\log_{2} (3)}{2} + \frac{5 \log_{2} (r)}{2} + \frac{1}{2} (- 3 \log_{2} (z))$

Step 9.3

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

Tap for more steps...

Step 9.3.1

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

$\frac{\log_{2} (3)}{2} + \frac{5 \log_{2} (r)}{2} + \frac{- 3}{2} \log_{2} (z)$

Step 9.3.2

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

$\frac{\log_{2} (3)}{2} + \frac{5 \log_{2} (r)}{2} + \frac{- 3 \log_{2} (z)}{2}$

Step 10

Move the negative in front of the fraction.

$\frac{\log_{2} (3)}{2} + \frac{5 \log_{2} (r)}{2} - \frac{3 \log_{2} (z)}{2}$