Precalculus Examples

$\ln (\sqrt{\frac{x y}{e^{7} z}})$

Step 1

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

$\ln ({(\frac{x y}{e^{7} z})}^{\frac{1}{2}})$

Step 2

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

$\frac{1}{2} \ln (\frac{x y}{e^{7} z})$

Step 3

Rewrite $\ln (\frac{x y}{e^{7} z})$ as $\ln (x y) - \ln (e^{7} z)$ .

$\frac{1}{2} (\ln (x y) - \ln (e^{7} z))$

Step 4

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

$\frac{1}{2} (\ln (x) + \ln (y) - \ln (e^{7} z))$

Step 5

Rewrite $\ln (e^{7} z)$ as $\ln (e^{7}) + \ln (z)$ .

$\frac{1}{2} (\ln (x) + \ln (y) - (\ln (e^{7}) + \ln (z)))$

Step 6

Expand $\ln (e^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{2} (\ln (x) + \ln (y) - (7 \ln (e) + \ln (z)))$

Step 7

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

$\frac{1}{2} (\ln (x) + \ln (y) - (7 \cdot 1 + \ln (z)))$

Step 8

Multiply $7$ by $1$ .

$\frac{1}{2} (\ln (x) + \ln (y) - (7 + \ln (z)))$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$\frac{1}{2} (\ln (x) + \ln (y) - 1 \cdot 7 - \ln (z))$

Step 9.2

Multiply $- 1$ by $7$ .

$\frac{1}{2} (\ln (x) + \ln (y) - 7 - \ln (z))$

Step 10

Apply the distributive property.

$\frac{1}{2} \ln (x) + \frac{1}{2} \ln (y) + \frac{1}{2} \cdot - 7 + \frac{1}{2} (- \ln (z))$

Step 11

Simplify.

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Step 11.1

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

$\frac{\ln (x)}{2} + \frac{1}{2} \ln (y) + \frac{1}{2} \cdot - 7 + \frac{1}{2} (- \ln (z))$

Step 11.2

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

$\frac{\ln (x)}{2} + \frac{\ln (y)}{2} + \frac{1}{2} \cdot - 7 + \frac{1}{2} (- \ln (z))$

Step 11.3

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

$\frac{\ln (x)}{2} + \frac{\ln (y)}{2} + \frac{- 7}{2} + \frac{1}{2} (- \ln (z))$

Step 11.4

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

$\frac{\ln (x)}{2} + \frac{\ln (y)}{2} + \frac{- 7}{2} - \frac{\ln (z)}{2}$

Step 12

Move the negative in front of the fraction.

$\frac{\ln (x)}{2} + \frac{\ln (y)}{2} - \frac{7}{2} - \frac{\ln (z)}{2}$