Calculus Examples

$\ln (\frac{4}{3}) + \ln (3 e) - \ln (4)$

Step 1

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

$\ln (\frac{4}{3} (3 e)) - \ln (4)$

Step 2

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

$\ln (\frac{\frac{4}{3} (3 e)}{4})$

Step 3

Simplify the numerator.

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Step 3.1

Combine $\frac{4}{3}$ and $3$ .

$\ln (\frac{\frac{4 \cdot 3}{3} e}{4})$

Step 3.2

Combine $\frac{4 \cdot 3}{3}$ and $e$ .

$\ln (\frac{\frac{4 \cdot 3 e}{3}}{4})$

Step 4

Multiply $4$ by $3$ .

$\ln (\frac{\frac{12 e}{3}}{4})$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Reduce the expression $\frac{12 e}{3}$ by cancelling the common factors.

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Step 5.1.1

Factor $3$ out of $12 e$ .

$\ln (\frac{\frac{3 (4 e)}{3}}{4})$

Step 5.1.2

Factor $3$ out of $3$ .

$\ln (\frac{\frac{3 (4 e)}{3 (1)}}{4})$

Step 5.1.3

Cancel the common factor.

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

Step 5.1.4

Rewrite the expression.

$\ln (\frac{\frac{4 e}{1}}{4})$

Step 5.2

Divide $4 e$ by $1$ .

$\ln (\frac{4 e}{4})$

Step 6

Cancel the common factor of $4$ .

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Step 6.1

Cancel the common factor.

$\ln (\frac{4 e}{4})$

Step 6.2

Divide $e$ by $1$ .

$\ln (e)$

Step 7

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

$1$