Precalculus Examples

$\log_{5} (\ln (\ln (e^{25})))$

Step 1

Expand $\ln (e^{25})$ by moving $25$ outside the logarithm.

$\log_{5} (\ln (25 \ln (e)))$

Step 2

Rewrite $\ln (25 \ln (e))$ as $\ln (25) + \ln (\ln (e))$ .

$\log_{5} (\ln (25) + \ln (\ln (e)))$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Rewrite $\ln (25)$ as $\ln (5^{2})$ .

$\log_{5} (\ln (5^{2}) + \ln (\ln (e)))$

Step 3.2

Expand $\ln (5^{2})$ by moving $2$ outside the logarithm.

$\log_{5} (2 \ln (5) + \ln (\ln (e)))$

Step 3.3

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

$\log_{5} (2 \ln (5) + \ln (1))$

Step 3.4

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

$\log_{5} (2 \ln (5) + 0)$

Step 4

Add $2 \ln (5)$ and $0$ .

$\log_{5} (2 \ln (5))$

Step 5

Rewrite $\log_{5} (2 \ln (5))$ as $\log_{5} (2) + \log_{5} (\ln (5))$ .

$\log_{5} (2) + \log_{5} (\ln (5))$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\log_{5} (2) + \log_{5} (\ln (5))$

Decimal Form:

$0.72636053 \dots$