Precalculus Examples

$\log (\sqrt{\frac{y}{z^{4} {(x + 3)}^{9}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{y}{z^{4} {(x + 3)}^{9}}}$ as ${(\frac{y}{z^{4} {(x + 3)}^{9}})}^{\frac{1}{2}}$ .

$\log ({(\frac{y}{z^{4} {(x + 3)}^{9}})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{y}{z^{4} {(x + 3)}^{9}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{y}{z^{4} {(x + 3)}^{9}})$

Step 3

Rewrite $\log (\frac{y}{z^{4} {(x + 3)}^{9}})$ as $\log (y) - \log (z^{4} {(x + 3)}^{9})$ .

$\frac{1}{2} (\log (y) - \log (z^{4} {(x + 3)}^{9}))$

Step 4

Rewrite $\log (z^{4} {(x + 3)}^{9})$ as $\log (z^{4}) + \log ({(x + 3)}^{9})$ .

$\frac{1}{2} (\log (y) - (\log (z^{4}) + \log ({(x + 3)}^{9})))$

Step 5

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

$\frac{1}{2} (\log (y) - (4 \log (z) + \log ({(x + 3)}^{9})))$

Step 6

Expand $\log ({(x + 3)}^{9})$ by moving $9$ outside the logarithm.

$\frac{1}{2} (\log (y) - (4 \log (z) + 9 \log (x + 3)))$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{1}{2} (\log (y) - (4 \log (z)) - (9 \log (x + 3)))$

Step 7.2

Multiply $4$ by $- 1$ .

$\frac{1}{2} (\log (y) - 4 \log (z) - (9 \log (x + 3)))$

Step 7.3

Multiply $9$ by $- 1$ .

$\frac{1}{2} (\log (y) - 4 \log (z) - 9 \log (x + 3))$

Step 8

Apply the distributive property.

$\frac{1}{2} \log (y) + \frac{1}{2} (- 4 \log (z)) + \frac{1}{2} (- 9 \log (x + 3))$

Step 9

Simplify.

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Step 9.1

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

$\frac{\log (y)}{2} + \frac{1}{2} (- 4 \log (z)) + \frac{1}{2} (- 9 \log (x + 3))$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

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

$\frac{\log (y)}{2} + \frac{1}{2} (2 (- 2 \log (z))) + \frac{1}{2} (- 9 \log (x + 3))$

Step 9.2.2

Cancel the common factor.

$\frac{\log (y)}{2} + \frac{1}{2} (2 (- 2 \log (z))) + \frac{1}{2} (- 9 \log (x + 3))$

Step 9.2.3

Rewrite the expression.

$\frac{\log (y)}{2} - 2 \log (z) + \frac{1}{2} (- 9 \log (x + 3))$

Step 9.3

Multiply $\frac{1}{2} (- 9 \log (x + 3))$ .

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Step 9.3.1

Combine $- 9$ and $\frac{1}{2}$ .

$\frac{\log (y)}{2} - 2 \log (z) + \frac{- 9}{2} \log (x + 3)$

Step 9.3.2

Combine $\frac{- 9}{2}$ and $\log (x + 3)$ .

$\frac{\log (y)}{2} - 2 \log (z) + \frac{- 9 \log (x + 3)}{2}$

Step 10

Move the negative in front of the fraction.

$\frac{\log (y)}{2} - 2 \log (z) - \frac{9 \log (x + 3)}{2}$

Step 11

Combine the numerators over the common denominator.

$\frac{\log (y) - 9 \log (x + 3)}{2} - 2 \log (z)$