Calculus Examples

$3 \log (2) + \frac{1}{2} \cdot \log (25) - \log (20)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (2)$ by moving $3$ inside the logarithm.

$\log (2^{3}) + \frac{1}{2} \cdot \log (25) - \log (20)$

Step 1.2

Raise $2$ to the power of $3$ .

$\log (8) + \frac{1}{2} \cdot \log (25) - \log (20)$

Step 1.3

Simplify $\frac{1}{2} \log (25)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (8) + \log (25^{\frac{1}{2}}) - \log (20)$

Step 1.4

Rewrite $25$ as $5^{2}$ .

$\log (8) + \log ({(5^{2})}^{\frac{1}{2}}) - \log (20)$

Step 1.5

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (8) + \log (5^{2 (\frac{1}{2})}) - \log (20)$

Step 1.6

Cancel the common factor of $2$ .

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Step 1.6.1

Cancel the common factor.

$\log (8) + \log (5^{2 (\frac{1}{2})}) - \log (20)$

Step 1.6.2

Rewrite the expression.

$\log (8) + \log (5^{1}) - \log (20)$

Step 1.7

Evaluate the exponent.

$\log (8) + \log (5) - \log (20)$

Step 2

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

$\log (8 \cdot 5) - \log (20)$

Step 3

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

$\log (\frac{8 \cdot 5}{20})$

Step 4

Cancel the common factor of $8$ and $20$ .

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Step 4.1

Factor $4$ out of $8 \cdot 5$ .

$\log (\frac{4 (2 \cdot 5)}{20})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $4$ out of $20$ .

$\log (\frac{4 (2 \cdot 5)}{4 (5)})$

Step 4.2.2

Cancel the common factor.

$\log (\frac{4 (2 \cdot 5)}{4 \cdot 5})$

Step 4.2.3

Rewrite the expression.

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

Step 5

Cancel the common factor of $5$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $2$ by $1$ .

$\log (2)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\log (2)$

Decimal Form:

$0.30102999 \dots$