Precalculus Examples

$\ln (\frac{x}{\sqrt[3]{x^{2 - 3}}})$

Step 1

Rewrite $\ln (\frac{x}{\sqrt[3]{x^{2 - 3}}})$ as $\ln (x) - \ln (\sqrt[3]{x^{2 - 3}})$ .

$\ln (x) - \ln (\sqrt[3]{x^{2 - 3}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2 - 3}}$ as $x^{\frac{2 - 3}{3}}$ .

$\ln (x) - \ln (x^{\frac{2 - 3}{3}})$

Step 3

Subtract $3$ from $2$ .

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

Step 4

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

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

$\ln (x) - - \frac{\ln (x)}{3}$

Step 7

Multiply $- 1$ by $- 1$ .

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

Step 8

Multiply $\frac{\ln (x)}{3}$ by $1$ .

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

Step 9

To write $\ln (x)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10

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

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Simplify the numerator.

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Step 12.1

Move $3$ to the left of $\ln (x)$ .

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

Step 12.2

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

$\frac{4 \ln (x)}{3}$