Precalculus Examples

$\log (\sqrt{\frac{x^{8}}{y^{7} z^{9}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x^{8}}{y^{7} z^{9}}}$ as ${(\frac{x^{8}}{y^{7} z^{9}})}^{\frac{1}{2}}$ .

$\log ({(\frac{x^{8}}{y^{7} z^{9}})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{x^{8}}{y^{7} z^{9}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{x^{8}}{y^{7} z^{9}})$

Step 3

Rewrite $\log (\frac{x^{8}}{y^{7} z^{9}})$ as $\log (x^{8}) - \log (y^{7} z^{9})$ .

$\frac{1}{2} (\log (x^{8}) - \log (y^{7} z^{9}))$

Step 4

Expand $\log (x^{8})$ by moving $8$ outside the logarithm.

$\frac{1}{2} (8 \log (x) - \log (y^{7} z^{9}))$

Step 5

Rewrite $\log (y^{7} z^{9})$ as $\log (y^{7}) + \log (z^{9})$ .

$\frac{1}{2} (8 \log (x) - (\log (y^{7}) + \log (z^{9})))$

Step 6

Expand $\log (y^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{2} (8 \log (x) - (7 \log (y) + \log (z^{9})))$

Step 7

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

$\frac{1}{2} (8 \log (x) - (7 \log (y) + 9 \log (z)))$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\frac{1}{2} (8 \log (x) - (7 \log (y)) - (9 \log (z)))$

Step 8.2

Multiply $7$ by $- 1$ .

$\frac{1}{2} (8 \log (x) - 7 \log (y) - (9 \log (z)))$

Step 8.3

Multiply $9$ by $- 1$ .

$\frac{1}{2} (8 \log (x) - 7 \log (y) - 9 \log (z))$

Step 9

Apply the distributive property.

$\frac{1}{2} (8 \log (x)) + \frac{1}{2} (- 7 \log (y)) + \frac{1}{2} (- 9 \log (z))$

Step 10

Simplify.

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Step 10.1

Cancel the common factor of $2$ .

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Step 10.1.1

Factor $2$ out of $8 \log (x)$ .

$\frac{1}{2} (2 (4 \log (x))) + \frac{1}{2} (- 7 \log (y)) + \frac{1}{2} (- 9 \log (z))$

Step 10.1.2

Cancel the common factor.

$\frac{1}{2} (2 (4 \log (x))) + \frac{1}{2} (- 7 \log (y)) + \frac{1}{2} (- 9 \log (z))$

Step 10.1.3

Rewrite the expression.

$4 \log (x) + \frac{1}{2} (- 7 \log (y)) + \frac{1}{2} (- 9 \log (z))$

Step 10.2

Multiply $\frac{1}{2} (- 7 \log (y))$ .

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Step 10.2.1

Combine $- 7$ and $\frac{1}{2}$ .

$4 \log (x) + \frac{- 7}{2} \log (y) + \frac{1}{2} (- 9 \log (z))$

Step 10.2.2

Combine $\frac{- 7}{2}$ and $\log (y)$ .

$4 \log (x) + \frac{- 7 \log (y)}{2} + \frac{1}{2} (- 9 \log (z))$

Step 10.3

Multiply $\frac{1}{2} (- 9 \log (z))$ .

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Step 10.3.1

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

$4 \log (x) + \frac{- 7 \log (y)}{2} + \frac{- 9}{2} \log (z)$

Step 10.3.2

Combine $\frac{- 9}{2}$ and $\log (z)$ .

$4 \log (x) + \frac{- 7 \log (y)}{2} + \frac{- 9 \log (z)}{2}$

Step 11

Simplify each term.

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Step 11.1

Move the negative in front of the fraction.

$4 \log (x) - \frac{7 \log (y)}{2} + \frac{- 9 \log (z)}{2}$

Step 11.2

Move the negative in front of the fraction.

$4 \log (x) - \frac{7 \log (y)}{2} - \frac{9 \log (z)}{2}$