Precalculus Examples

$2 \log_{2} (25) - 2 \log_{2} (5) + \log_{2} (3)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{2} (25)$ by moving $2$ inside the logarithm.

$\log_{2} (25^{2}) - 2 \log_{2} (5) + \log_{2} (3)$

Step 1.2

Raise $25$ to the power of $2$ .

$\log_{2} (625) - 2 \log_{2} (5) + \log_{2} (3)$

Step 1.3

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

$\log_{2} (625) - \log_{2} (5^{2}) + \log_{2} (3)$

Step 1.4

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

$\log_{2} (625) - \log_{2} (25) + \log_{2} (3)$

Step 2

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

$\log_{2} (\frac{625}{25}) + \log_{2} (3)$

Step 3

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

$\log_{2} (\frac{625}{25} \cdot 3)$

Step 4

Divide $625$ by $25$ .

$\log_{2} (25 \cdot 3)$

Step 5

Multiply $25$ by $3$ .

$\log_{2} (75)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\log_{2} (75)$

Decimal Form:

$6.22881869 \dots$