Precalculus Examples

$\log (x^{19} \sqrt{\frac{y^{13}}{z^{19}}})$

Step 1

Rewrite $\log (x^{19} \sqrt{\frac{y^{13}}{z^{19}}})$ as $\log (x^{19}) + \log (\sqrt{\frac{y^{13}}{z^{19}}})$ .

$\log (x^{19}) + \log (\sqrt{\frac{y^{13}}{z^{19}}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{y^{13}}{z^{19}}}$ as ${(\frac{y^{13}}{z^{19}})}^{\frac{1}{2}}$ .

$\log (x^{19}) + \log ({(\frac{y^{13}}{z^{19}})}^{\frac{1}{2}})$

Step 3

Expand $\log (x^{19})$ by moving $19$ outside the logarithm.

$19 \log (x) + \log ({(\frac{y^{13}}{z^{19}})}^{\frac{1}{2}})$

Step 4

Expand $\log ({(\frac{y^{13}}{z^{19}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$19 \log (x) + \frac{1}{2} \log (\frac{y^{13}}{z^{19}})$

Step 5

Rewrite $\log (\frac{y^{13}}{z^{19}})$ as $\log (y^{13}) - \log (z^{19})$ .

$19 \log (x) + \frac{1}{2} (\log (y^{13}) - \log (z^{19}))$

Step 6

Expand $\log (y^{13})$ by moving $13$ outside the logarithm.

$19 \log (x) + \frac{1}{2} (13 \log (y) - \log (z^{19}))$

Step 7

Expand $\log (z^{19})$ by moving $19$ outside the logarithm.

$19 \log (x) + \frac{1}{2} (13 \log (y) - (19 \log (z)))$

Step 8

Multiply $19$ by $- 1$ .

$19 \log (x) + \frac{1}{2} (13 \log (y) - 19 \log (z))$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$19 \log (x) + \frac{1}{2} (13 \log (y)) + \frac{1}{2} (- 19 \log (z))$

Step 9.2

Multiply $\frac{1}{2} (13 \log (y))$ .

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Step 9.2.1

Combine $13$ and $\frac{1}{2}$ .

$19 \log (x) + \frac{13}{2} \log (y) + \frac{1}{2} (- 19 \log (z))$

Step 9.2.2

Combine $\frac{13}{2}$ and $\log (y)$ .

$19 \log (x) + \frac{13 \log (y)}{2} + \frac{1}{2} (- 19 \log (z))$

Step 9.3

Multiply $\frac{1}{2} (- 19 \log (z))$ .

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Step 9.3.1

Combine $- 19$ and $\frac{1}{2}$ .

$19 \log (x) + \frac{13 \log (y)}{2} + \frac{- 19}{2} \log (z)$

Step 9.3.2

Combine $\frac{- 19}{2}$ and $\log (z)$ .

$19 \log (x) + \frac{13 \log (y)}{2} + \frac{- 19 \log (z)}{2}$

Step 9.4

Move the negative in front of the fraction.

$19 \log (x) + \frac{13 \log (y)}{2} - \frac{19 \log (z)}{2}$