Precalculus Examples

$\log (\sqrt{\frac{x^{3} y}{z^{4}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x^{3} y}{z^{4}}}$ as ${(\frac{x^{3} y}{z^{4}})}^{\frac{1}{2}}$ .

$\log ({(\frac{x^{3} y}{z^{4}})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{x^{3} y}{z^{4}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{x^{3} y}{z^{4}})$

Step 3

Rewrite $\log (\frac{x^{3} y}{z^{4}})$ as $\log (x^{3} y) - \log (z^{4})$ .

$\frac{1}{2} (\log (x^{3} y) - \log (z^{4}))$

Step 4

Expand $\log (z^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{2} (\log (x^{3} y) - (4 \log (z)))$

Step 5

Multiply $4$ by $- 1$ .

$\frac{1}{2} (\log (x^{3} y) - 4 \log (z))$

Step 6

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

$\frac{1}{2} (\log (x^{3}) + \log (y) - 4 \log (z))$

Step 7

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

$\frac{1}{2} (3 \log (x) + \log (y) - 4 \log (z))$

Step 8

Apply the distributive property.

$\frac{1}{2} (3 \log (x)) + \frac{1}{2} \log (y) + \frac{1}{2} (- 4 \log (z))$

Step 9

Simplify.

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Step 9.1

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

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Step 9.1.1

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

$\frac{3}{2} \log (x) + \frac{1}{2} \log (y) + \frac{1}{2} (- 4 \log (z))$

Step 9.1.2

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

$\frac{3 \log (x)}{2} + \frac{1}{2} \log (y) + \frac{1}{2} (- 4 \log (z))$

Step 9.2

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

$\frac{3 \log (x)}{2} + \frac{\log (y)}{2} + \frac{1}{2} (- 4 \log (z))$

Step 9.3

Cancel the common factor of $2$ .

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Step 9.3.1

Factor $2$ out of $- 4 \log (z)$ .

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

Step 9.3.2

Cancel the common factor.

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

Step 9.3.3

Rewrite the expression.

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