Precalculus Examples

$\ln (10) - \ln (5) - \frac{1}{3} \cdot \ln (8)$

Step 1

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

$\ln (\frac{10}{5}) - \frac{1}{3} \cdot \ln (8)$

Step 2

Simplify each term.

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Step 2.1

Divide $10$ by $5$ .

$\ln (2) - \frac{1}{3} \cdot \ln (8)$

Step 2.2

Multiply $- \frac{1}{3} \ln (8)$ .

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Step 2.2.1

Reorder $\ln (8)$ and $\frac{1}{3}$ .

$\ln (2) - (\frac{1}{3} \ln (8))$

Step 2.2.2

Simplify $\frac{1}{3} \ln (8)$ by moving $\frac{1}{3}$ inside the logarithm.

$\ln (2) - \ln (8^{\frac{1}{3}})$

Step 2.3

Rewrite $8$ as $2^{3}$ .

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

Step 2.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

$\ln (2) - \ln (2^{1})$

Step 2.6

Evaluate the exponent.

$\ln (2) - \ln (2)$

Step 3

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

$\ln (\frac{2}{2})$

Step 4

Divide $2$ by $2$ .

$\ln (1)$

Step 5

The natural logarithm of $1$ is $0$ .

$0$