Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\log (\frac{3 x^{4} y}{w z^{2}})$ as $\log (3 x^{4} y) - \log (w z^{2})$ .

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

Step 4

Rewrite $\log (3 x^{4} y)$ as $\log (3 x^{4}) + \log (y)$ .

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

Step 5

Rewrite $\log (3 x^{4})$ as $\log (3) + \log (x^{4})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $2$ by $- 1$ .

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

Step 10

Apply the distributive property.

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Cancel the common factor of $2$ .

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Step 11.2.1

Factor $2$ out of $4 \log (x)$ .

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

Step 11.2.2

Cancel the common factor.

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

Step 11.2.3

Rewrite the expression.

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

Step 11.3

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

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

Step 11.4

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

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

Step 11.5

Cancel the common factor of $2$ .

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Step 11.5.1

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

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

Step 11.5.2

Cancel the common factor.

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

Step 11.5.3

Rewrite the expression.

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