Precalculus Examples

$\log_{2} (6) + 3 \log_{2} (3) - 2 \log_{2} (5) - \log_{2} (7)$

Step 1

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

$\log_{2} (\frac{6}{7}) + 3 \log_{2} (3) - 2 \log_{2} (5)$

Step 2

Simplify each term.

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Step 2.1

Simplify $3 \log_{2} (3)$ by moving $3$ inside the logarithm.

$\log_{2} (\frac{6}{7}) + \log_{2} (3^{3}) - 2 \log_{2} (5)$

Step 2.2

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

$\log_{2} (\frac{6}{7}) + \log_{2} (27) - 2 \log_{2} (5)$

Step 2.3

Simplify $- 2 \log_{2} (5)$ by moving $2$ inside the logarithm.

$\log_{2} (\frac{6}{7}) + \log_{2} (27) - \log_{2} (5^{2})$

Step 2.4

Raise $5$ to the power of $2$ .

$\log_{2} (\frac{6}{7}) + \log_{2} (27) - \log_{2} (25)$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{2} (\frac{6}{7} \cdot 27) - \log_{2} (25)$

Step 4

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

$\log_{2} (\frac{\frac{6}{7} \cdot 27}{25})$

Step 5

Combine $\frac{6}{7}$ and $27$ .

$\log_{2} (\frac{\frac{6 \cdot 27}{7}}{25})$

Step 6

Multiply $6$ by $27$ .

$\log_{2} (\frac{\frac{162}{7}}{25})$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\log_{2} (\frac{162}{7} \cdot \frac{1}{25})$

Step 8

Multiply $\frac{162}{7} \cdot \frac{1}{25}$ .

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Step 8.1

Multiply $\frac{162}{7}$ by $\frac{1}{25}$ .

$\log_{2} (\frac{162}{7 \cdot 25})$

Step 8.2

Multiply $7$ by $25$ .

$\log_{2} (\frac{162}{175})$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\log_{2} (\frac{162}{175})$

Decimal Form:

$- 0.11136110 \dots$