Calculus Examples

$\frac{2}{3} - 2 \ln (2) - (\frac{1}{3} - \ln (1))$

Step 1

Find the common denominator.

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Step 1.1

Write $- 2 \ln (2)$ as a fraction with denominator $1$ .

$\frac{2}{3} + \frac{- 2 \ln (2)}{1} - (\frac{1}{3} - \ln (1))$

Step 1.2

Multiply $\frac{- 2 \ln (2)}{1}$ by $\frac{3}{3}$ .

$\frac{2}{3} + \frac{- 2 \ln (2)}{1} \cdot \frac{3}{3} - (\frac{1}{3} - \ln (1))$

Step 1.3

Multiply $\frac{- 2 \ln (2)}{1}$ by $\frac{3}{3}$ .

$\frac{2}{3} + \frac{- 2 \ln (2) \cdot 3}{3} - (\frac{1}{3} - \ln (1))$

Step 1.4

Write $- (\frac{1}{3} - \ln (1))$ as a fraction with denominator $1$ .

$\frac{2}{3} + \frac{- 2 \ln (2) \cdot 3}{3} + \frac{- (\frac{1}{3} - \ln (1))}{1}$

Step 1.5

Multiply $\frac{- (\frac{1}{3} - \ln (1))}{1}$ by $\frac{3}{3}$ .

$\frac{2}{3} + \frac{- 2 \ln (2) \cdot 3}{3} + \frac{- (\frac{1}{3} - \ln (1))}{1} \cdot \frac{3}{3}$

Step 1.6

Multiply $\frac{- (\frac{1}{3} - \ln (1))}{1}$ by $\frac{3}{3}$ .

$\frac{2}{3} + \frac{- 2 \ln (2) \cdot 3}{3} + \frac{- (\frac{1}{3} - \ln (1)) \cdot 3}{3}$

Step 2

Combine the numerators over the common denominator.

$\frac{2 - 2 \ln (2) \cdot 3 - (\frac{1}{3} - \ln (1)) \cdot 3}{3}$

Step 3

Simplify each term.

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Step 3.1

Simplify $- 2 \ln (2)$ by moving $2$ inside the logarithm.

$\frac{2 - \ln (2^{2}) \cdot 3 - (\frac{1}{3} - \ln (1)) \cdot 3}{3}$

Step 3.2

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

$\frac{2 - \ln (4) \cdot 3 - (\frac{1}{3} - \ln (1)) \cdot 3}{3}$

Step 3.3

Multiply $- \ln (4) \cdot 3$ .

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Step 3.3.1

Multiply $3$ by $- 1$ .

$\frac{2 - 3 \ln (4) - (\frac{1}{3} - \ln (1)) \cdot 3}{3}$

Step 3.3.2

Simplify $- 3 \ln (4)$ by moving $3$ inside the logarithm.

$\frac{2 - \ln (4^{3}) - (\frac{1}{3} - \ln (1)) \cdot 3}{3}$

Step 3.4

Raise $4$ to the power of $3$ .

$\frac{2 - \ln (64) - (\frac{1}{3} - \ln (1)) \cdot 3}{3}$

Step 3.5

Simplify each term.

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Step 3.5.1

The natural logarithm of $1$ is $0$ .

$\frac{2 - \ln (64) - (\frac{1}{3} - 0) \cdot 3}{3}$

Step 3.5.2

Multiply $- 1$ by $0$ .

$\frac{2 - \ln (64) - (\frac{1}{3} + 0) \cdot 3}{3}$

Step 3.6

Add $\frac{1}{3}$ and $0$ .

$\frac{2 - \ln (64) - \frac{1}{3} \cdot 3}{3}$

Step 3.7

Cancel the common factor of $3$ .

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Step 3.7.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{2 - \ln (64) + \frac{- 1}{3} \cdot 3}{3}$

Step 3.7.2

Cancel the common factor.

$\frac{2 - \ln (64) + \frac{- 1}{3} \cdot 3}{3}$

Step 3.7.3

Rewrite the expression.

$\frac{2 - \ln (64) - 1}{3}$

Step 4

Subtract $1$ from $2$ .

$\frac{1 - \ln (64)}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1 - \ln (64)}{3}$

Decimal Form:

$- 1.05296102 \dots$