Precalculus Examples

$\ln (\frac{\sqrt{x y}}{e^{3} z})$

Step 1

Rewrite $\ln (\frac{\sqrt{x y}}{e^{3} z})$ as $\ln (\sqrt{x y}) - \ln (e^{3} z)$ .

$\ln (\sqrt{x y}) - \ln (e^{3} z)$

Step 2

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

$\ln ({(x y)}^{\frac{1}{2}}) - \ln (e^{3} z)$

Step 3

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

$\frac{1}{2} \ln (x y) - \ln (e^{3} z)$

Step 4

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

$\frac{1}{2} (\ln (x) + \ln (y)) - \ln (e^{3} z)$

Step 5

Rewrite $\ln (e^{3} z)$ as $\ln (e^{3}) + \ln (z)$ .

$\frac{1}{2} (\ln (x) + \ln (y)) - (\ln (e^{3}) + \ln (z))$

Step 6

Expand $\ln (e^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (\ln (x) + \ln (y)) - (3 \ln (e) + \ln (z))$

Step 7

The natural logarithm of $e$ is $1$ .

$\frac{1}{2} (\ln (x) + \ln (y)) - (3 \cdot 1 + \ln (z))$

Step 8

Multiply $3$ by $1$ .

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

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Apply the distributive property.

$\frac{\ln (x)}{2} + \frac{\ln (y)}{2} - 1 \cdot 3 - \ln (z)$

Step 9.5

Multiply $- 1$ by $3$ .

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