Precalculus Examples

$\ln (e \cdot \sqrt{\frac{x^{6}}{e^{2}}})$

Step 1

Rewrite $\ln (e \cdot \sqrt{\frac{x^{6}}{e^{2}}})$ as $\ln (e) + \ln (\sqrt{\frac{x^{6}}{e^{2}}})$ .

$\ln (e) + \ln (\sqrt{\frac{x^{6}}{e^{2}}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x^{6}}{e^{2}}}$ as ${(\frac{x^{6}}{e^{2}})}^{\frac{1}{2}}$ .

$\ln (e) + \ln ({(\frac{x^{6}}{e^{2}})}^{\frac{1}{2}})$

Step 3

Expand $\ln ({(\frac{x^{6}}{e^{2}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\ln (e) + \frac{1}{2} \ln (\frac{x^{6}}{e^{2}})$

Step 4

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

$1 + \frac{1}{2} \ln (\frac{x^{6}}{e^{2}})$

Step 5

Rewrite $\ln (\frac{x^{6}}{e^{2}})$ as $\ln (x^{6}) - \ln (e^{2})$ .

$1 + \frac{1}{2} (\ln (x^{6}) - \ln (e^{2}))$

Step 6

Expand $\ln (x^{6})$ by moving $6$ outside the logarithm.

$1 + \frac{1}{2} (6 \ln (x) - \ln (e^{2}))$

Step 7

Expand $\ln (e^{2})$ by moving $2$ outside the logarithm.

$1 + \frac{1}{2} (6 \ln (x) - (2 \ln (e)))$

Step 8

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

$1 + \frac{1}{2} (6 \ln (x) - (2 \cdot 1))$

Step 9

Multiply $2$ by $1$ .

$1 + \frac{1}{2} (6 \ln (x) - 1 \cdot 2)$

Step 10

Multiply $- 1$ by $2$ .

$1 + \frac{1}{2} (6 \ln (x) - 2)$

Step 11

Simplify each term.

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Step 11.1

Apply the distributive property.

$1 + \frac{1}{2} (6 \ln (x)) + \frac{1}{2} \cdot - 2$

Step 11.2

Cancel the common factor of $2$ .

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Step 11.2.1

Factor $2$ out of $6 \ln (x)$ .

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

Step 11.2.2

Cancel the common factor.

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

Step 11.2.3

Rewrite the expression.

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

Step 11.3

Cancel the common factor of $2$ .

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Step 11.3.1

Factor $2$ out of $- 2$ .

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

Step 11.3.2

Cancel the common factor.

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

Step 11.3.3

Rewrite the expression.

$1 + 3 \ln (x) - 1$

Step 12

Subtract $1$ from $1$ .

$3 \ln (x) + 0$

Step 13

Add $3 \ln (x)$ and $0$ .

$3 \ln (x)$