Precalculus Examples

$\log (\frac{\sqrt{x z}}{z^{5}})$

Step 1

Rewrite $\log (\frac{\sqrt{x z}}{z^{5}})$ as $\log (\sqrt{x z}) - \log (z^{5})$ .

$\log (\sqrt{x z}) - \log (z^{5})$

Step 2

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

$\log ({(x z)}^{\frac{1}{2}}) - \log (z^{5})$

Step 3

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

$\frac{1}{2} \log (x z) - \log (z^{5})$

Step 4

Expand $\log (z^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} \log (x z) - (5 \log (z))$

Step 5

Multiply $5$ by $- 1$ .

$\frac{1}{2} \log (x z) - 5 \log (z)$

Step 6

Rewrite $\log (x z)$ as $\log (x) + \log (z)$ .

$\frac{1}{2} (\log (x) + \log (z)) - 5 \log (z)$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{1}{2} \log (x) + \frac{1}{2} \log (z) - 5 \log (z)$

Step 7.2

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

$\frac{\log (x)}{2} + \frac{1}{2} \log (z) - 5 \log (z)$

Step 7.3

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

$\frac{\log (x)}{2} + \frac{\log (z)}{2} - 5 \log (z)$

Step 8

To write $- 5 \log (z)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\log (x)}{2} + \frac{\log (z)}{2} - 5 \log (z) \cdot \frac{2}{2}$

Step 9

Combine $- 5 \log (z)$ and $\frac{2}{2}$ .

$\frac{\log (x)}{2} + \frac{\log (z)}{2} + \frac{- 5 \log (z) \cdot 2}{2}$

Step 10

Combine the numerators over the common denominator.

$\frac{\log (x)}{2} + \frac{\log (z) - 5 \log (z) \cdot 2}{2}$

Step 11

Combine the numerators over the common denominator.

$\frac{\log (x) + \log (z) - 5 \log (z) \cdot 2}{2}$

Step 12

Simplify the numerator.

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Step 12.1

Multiply $2$ by $- 5$ .

$\frac{\log (x) + \log (z) - 10 \log (z)}{2}$

Step 12.2

Subtract $10 \log (z)$ from $\log (z)$ .

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