Precalculus Examples

$\ln (\sqrt{\frac{z}{x y}})$

Step 1

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

$\ln ({(\frac{z}{x y})}^{\frac{1}{2}})$

Step 2

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

$\frac{1}{2} \ln (\frac{z}{x y})$

Step 3

Rewrite $\ln (\frac{z}{x y})$ as $\ln (z) - \ln (x y)$ .

$\frac{1}{2} (\ln (z) - \ln (x y))$

Step 4

Rewrite $\ln (x y)$ as $\ln (x) + \ln (y)$ .

$\frac{1}{2} (\ln (z) - (\ln (x) + \ln (y)))$

Step 5

Apply the distributive property.

$\frac{1}{2} (\ln (z) - \ln (x) - \ln (y))$

Step 6

Apply the distributive property.

$\frac{1}{2} \ln (z) + \frac{1}{2} (- \ln (x)) + \frac{1}{2} (- \ln (y))$

Step 7

Simplify.

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

$\frac{\ln (z)}{2} + \frac{1}{2} (- \ln (x)) + \frac{1}{2} (- \ln (y))$

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

$\frac{\ln (z)}{2} - \frac{\ln (x)}{2} + \frac{1}{2} (- \ln (y))$

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

$\frac{\ln (z)}{2} - \frac{\ln (x)}{2} - \frac{\ln (y)}{2}$