Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $5$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Multiply $\frac{1}{2} (3 \log_{m} (r))$ .

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.3

Multiply $\frac{1}{2} (- 5 \log_{m} (z))$ .

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Step 9.3.1

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

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

Step 9.3.2

Combine $\frac{- 5}{2}$ and $\log_{m} (z)$ .

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

Step 10

Move the negative in front of the fraction.

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