Precalculus Examples

$\ln (\sqrt[3]{e^{3} x})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{e^{3} x}$ as ${(e^{3} x)}^{\frac{1}{3}}$ .

$\ln ({(e^{3} x)}^{\frac{1}{3}})$

Step 2

Expand $\ln ({(e^{3} x)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \ln (e^{3} x)$

Step 3

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

$\frac{1}{3} (\ln (e^{3}) + \ln (x))$

Step 4

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

$\frac{1}{3} (3 \ln (e) + \ln (x))$

Step 5

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

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

Step 6

Multiply $3$ by $1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Cancel the common factor of $3$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

Combine $\frac{1}{3}$ and $\ln (x)$ .

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