Calculus Examples

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

Step 1

Expand $\ln (e^{e^{7}})$ by moving $e^{7}$ outside the logarithm.

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

Step 2

Rewrite $\ln (e^{7} \ln (e))$ as $\ln (e^{7}) + \ln (\ln (e))$ .

$\ln (e^{7}) + \ln (\ln (e))$

Step 3

Use logarithm rules to move $7$ out of the exponent.

$7 \ln (e) + \ln (\ln (e))$

Step 4

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

$7 \cdot 1 + \ln (\ln (e))$

Step 5

Multiply $7$ by $1$ .

$7 + \ln (\ln (e))$

Step 6

Simplify each term.

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Step 6.1

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

$7 + \ln (1)$

Step 6.2

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

$7 + 0$

Step 7

Add $7$ and $0$ .

$7$