Precalculus Examples

$2 \log_{8} (7) + \log_{8} (4) - \log_{8} (2)$

Step 1

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

$2 \log_{8} (7) + \log_{8} (\frac{4}{2})$

Step 2

Simplify each term.

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Step 2.1

Simplify $2 \log_{8} (7)$ by moving $2$ inside the logarithm.

$\log_{8} (7^{2}) + \log_{8} (\frac{4}{2})$

Step 2.2

Raise $7$ to the power of $2$ .

$\log_{8} (49) + \log_{8} (\frac{4}{2})$

Step 2.3

Logarithm base $8$ of $\frac{4}{2}$ is $\frac{1}{3}$ .

$\log_{8} (49) + \frac{1}{3}$

Step 3

To write $\log_{8} (49)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log_{8} (49) \cdot \frac{3}{3} + \frac{1}{3}$

Step 4

Combine fractions.

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Step 4.1

Combine $\log_{8} (49)$ and $\frac{3}{3}$ .

$\frac{\log_{8} (49) \cdot 3}{3} + \frac{1}{3}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{\log_{8} (49) \cdot 3 + 1}{3}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $\log_{8} (49) \cdot 3$ .

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Step 5.1.1

Reorder $\log_{8} (49)$ and $3$ .

$\frac{3 \cdot \log_{8} (49) + 1}{3}$

Step 5.1.2

Simplify $3 \cdot \log_{8} (49)$ by moving $3$ inside the logarithm.

$\frac{\log_{8} (49^{3}) + 1}{3}$

Step 5.2

Raise $49$ to the power of $3$ .

$\frac{\log_{8} (117649) + 1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\log_{8} (117649) + 1}{3}$

Decimal Form:

$2.20490328 \dots$