Precalculus Examples

$\frac{1}{3} \cdot (\ln (4) - \ln (32))$

Step 1

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

$\frac{1}{3} \cdot \ln (\frac{4}{32})$

Step 2

Cancel the common factor of $4$ and $32$ .

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Step 2.1

Factor $4$ out of $4$ .

$\frac{1}{3} \cdot \ln (\frac{4 (1)}{32})$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $4$ out of $32$ .

$\frac{1}{3} \cdot \ln (\frac{4 \cdot 1}{4 \cdot 8})$

Step 2.2.2

Cancel the common factor.

$\frac{1}{3} \cdot \ln (\frac{4 \cdot 1}{4 \cdot 8})$

Step 2.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Simplify the expression.

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Step 4.1

Apply the product rule to $\frac{1}{8}$ .

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

Step 4.2

One to any power is one.

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 5.4

Evaluate the exponent.

$\ln (\frac{1}{2})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{1}{2})$

Decimal Form:

$- 0.69314718 \dots$