Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\log (\frac{2 x^{3}}{y z^{2}})$ as $\log (2 x^{3}) - \log (y z^{2})$ .

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

Step 4

Rewrite $\log (2 x^{3})$ as $\log (2) + \log (x^{3})$ .

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

Step 5

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

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

Step 6

Rewrite $\log (y z^{2})$ as $\log (y) + \log (z^{2})$ .

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

Step 7

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

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

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $2$ by $- 1$ .

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

Step 9

Apply the distributive property.

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

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

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Step 10.2.1

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

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

Step 10.2.2

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

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

Step 10.3

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

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

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

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

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

Step 10.4.2

Cancel the common factor.

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

Step 10.4.3

Rewrite the expression.

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