Algebra Examples

$\ln (e^{11}) - \ln (e^{7})$

Step 1

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

$\ln (\frac{e^{11}}{e^{7}})$

Step 2

Cancel the common factor of $e^{11}$ and $e^{7}$ .

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Step 2.1

Factor $e^{7}$ out of $e^{11}$ .

$\ln (\frac{e^{7} e^{4}}{e^{7}})$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Multiply by $1$ .

$\ln (\frac{e^{7} e^{4}}{e^{7} \cdot 1})$

Step 2.2.2

Cancel the common factor.

$\ln (\frac{e^{7} e^{4}}{e^{7} \cdot 1})$

Step 2.2.3

Rewrite the expression.

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

Step 2.2.4

Divide $e^{4}$ by $1$ .

$\ln (e^{4})$

Step 3

Use logarithm rules to move $4$ out of the exponent.

$4 \ln (e)$

Step 4

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

$4 \cdot 1$

Step 5

Multiply $4$ by $1$ .

$4$