Precalculus Examples

$\frac{\log_{3} (27) + 8 \log_{16} (2)}{\log_{8} (512)} \cdot \log_{1000} (10)$

Step 1

Simplify the numerator.

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Step 1.1

Logarithm base $3$ of $27$ is $3$ .

$\frac{3 + 8 \log_{16} (2)}{\log_{8} (512)} \cdot \log_{1000} (10)$

Step 1.2

Logarithm base $16$ of $2$ is $\frac{1}{4}$ .

$\frac{3 + 8 (\frac{1}{4})}{\log_{8} (512)} \cdot \log_{1000} (10)$

Step 1.3

Cancel the common factor of $4$ .

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Step 1.3.1

Factor $4$ out of $8$ .

$\frac{3 + 4 (2) \frac{1}{4}}{\log_{8} (512)} \cdot \log_{1000} (10)$

Step 1.3.2

Cancel the common factor.

$\frac{3 + 4 \cdot 2 \frac{1}{4}}{\log_{8} (512)} \cdot \log_{1000} (10)$

Step 1.3.3

Rewrite the expression.

$\frac{3 + 2}{\log_{8} (512)} \cdot \log_{1000} (10)$

Step 1.4

Add $3$ and $2$ .

$\frac{5}{\log_{8} (512)} \cdot \log_{1000} (10)$

Step 2

Logarithm base $8$ of $512$ is $3$ .

$\frac{5}{3} \cdot \log_{1000} (10)$

Step 3

Logarithm base $1000$ of $10$ is $\frac{1}{3}$ .

$\frac{5}{3} \cdot \frac{1}{3}$

Step 4

Multiply $\frac{5}{3} \cdot \frac{1}{3}$ .

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Step 4.1

Multiply $\frac{5}{3}$ by $\frac{1}{3}$ .

$\frac{5}{3 \cdot 3}$

Step 4.2

Multiply $3$ by $3$ .

$\frac{5}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{9}$

Decimal Form:

$0.\overline{5}$