Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{3} \ln (6)$ by moving $\frac{1}{3}$ inside the logarithm.

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

Step 1.2

Multiply $- \frac{1}{3} \ln (2)$ .

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Step 1.2.1

Reorder $\ln (2)$ and $\frac{1}{3}$ .

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

Step 1.2.2

Simplify $\frac{1}{3} \ln (2)$ by moving $\frac{1}{3}$ inside the logarithm.

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

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{6^{\frac{1}{3}}}{2^{\frac{1}{3}}})$

Step 3

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$\ln ({(\frac{6}{2})}^{\frac{1}{3}})$

Step 4

Divide $6$ by $2$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.36620409 \dots$