Precalculus Examples

$\ln (x^{6}) - 3 \ln (\sqrt[3]{x^{2}})$

Step 1

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

$6 \ln (x) - 3 \ln (\sqrt[3]{x^{2}})$

Step 2

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

$6 \ln (x) - 3 \ln (x^{\frac{2}{3}})$

Step 3

Expand $\ln (x^{\frac{2}{3}})$ by moving $\frac{2}{3}$ outside the logarithm.

$6 \ln (x) - 3 (\frac{2}{3} \ln (x))$

Step 4

Simplify each term.

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Step 4.1

Combine $\frac{2}{3}$ and $\ln (x)$ .

$6 \ln (x) - 3 \frac{2 \ln (x)}{3}$

Step 4.2

Cancel the common factor of $3$ .

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Step 4.2.1

Factor $3$ out of $- 3$ .

$6 \ln (x) + 3 (- 1) \frac{2 \ln (x)}{3}$

Step 4.2.2

Cancel the common factor.

$6 \ln (x) + 3 \cdot - 1 \frac{2 \ln (x)}{3}$

Step 4.2.3

Rewrite the expression.

$6 \ln (x) - 1 (2 \ln (x))$

Step 4.3

Multiply $2$ by $- 1$ .

$6 \ln (x) - 2 \ln (x)$

Step 5

Subtract $2 \ln (x)$ from $6 \ln (x)$ .

$4 \ln (x)$