Precalculus Examples

$\log (20) + \frac{1}{2} \cdot \log (2) - \log (5)$

Step 1

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

$\frac{1}{2} \cdot \log (2) + \log (\frac{20}{5})$

Step 2

Simplify each term.

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Step 2.1

Simplify $\frac{1}{2} \log (2)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (2^{\frac{1}{2}}) + \log (\frac{20}{5})$

Step 2.2

Divide $20$ by $5$ .

$\log (2^{\frac{1}{2}}) + \log (4)$

Step 3

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

$\log (2^{\frac{1}{2}} \cdot 4)$

Step 4

Multiply $2^{\frac{1}{2}} \cdot 4$ .

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Step 4.1

Rewrite $4$ as $2^{2}$ .

$\log (2^{\frac{1}{2}} \cdot 2^{2})$

Step 4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (2^{\frac{1}{2} + 2})$

Step 4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log (2^{\frac{1}{2} + 2 \cdot \frac{2}{2}})$

Step 4.4

Combine $2$ and $\frac{2}{2}$ .

$\log (2^{\frac{1}{2} + \frac{2 \cdot 2}{2}})$

Step 4.5

Combine the numerators over the common denominator.

$\log (2^{\frac{1 + 2 \cdot 2}{2}})$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $2$ by $2$ .

$\log (2^{\frac{1 + 4}{2}})$

Step 4.6.2

Add $1$ and $4$ .

$\log (2^{\frac{5}{2}})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\log (2^{\frac{5}{2}})$

Decimal Form:

$0.75257498 \dots$