Precalculus Examples

$\log (\sqrt{\frac{3 x y}{z^{3}}})$

Step 1

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

$\log ({(\frac{3 x y}{z^{3}})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{3 x y}{z^{3}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{3 x y}{z^{3}})$

Step 3

Rewrite $\log (\frac{3 x y}{z^{3}})$ as $\log (3 x y) - \log (z^{3})$ .

$\frac{1}{2} (\log (3 x y) - \log (z^{3}))$

Step 4

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

$\frac{1}{2} (\log (3 x y) - (3 \log (z)))$

Step 5

Multiply $3$ by $- 1$ .

$\frac{1}{2} (\log (3 x y) - 3 \log (z))$

Step 6

Rewrite $\log (3 x y)$ as $\log (3 x) + \log (y)$ .

$\frac{1}{2} (\log (3 x) + \log (y) - 3 \log (z))$

Step 7

Rewrite $\log (3 x)$ as $\log (3) + \log (x)$ .

$\frac{1}{2} (\log (3) + \log (x) + \log (y) - 3 \log (z))$

Step 8

Apply the distributive property.

$\frac{1}{2} \log (3) + \frac{1}{2} \log (x) + \frac{1}{2} \log (y) + \frac{1}{2} (- 3 \log (z))$

Step 9

Simplify.

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Step 9.1

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

$\frac{\log (3)}{2} + \frac{1}{2} \log (x) + \frac{1}{2} \log (y) + \frac{1}{2} (- 3 \log (z))$

Step 9.2

Combine $\frac{1}{2}$ and $\log (x)$ .

$\frac{\log (3)}{2} + \frac{\log (x)}{2} + \frac{1}{2} \log (y) + \frac{1}{2} (- 3 \log (z))$

Step 9.3

Combine $\frac{1}{2}$ and $\log (y)$ .

$\frac{\log (3)}{2} + \frac{\log (x)}{2} + \frac{\log (y)}{2} + \frac{1}{2} (- 3 \log (z))$

Step 9.4

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

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Step 9.4.1

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

$\frac{\log (3)}{2} + \frac{\log (x)}{2} + \frac{\log (y)}{2} + \frac{- 3}{2} \log (z)$

Step 9.4.2

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

$\frac{\log (3)}{2} + \frac{\log (x)}{2} + \frac{\log (y)}{2} + \frac{- 3 \log (z)}{2}$

Step 10

Move the negative in front of the fraction.

$\frac{\log (3)}{2} + \frac{\log (x)}{2} + \frac{\log (y)}{2} - \frac{3 \log (z)}{2}$