Precalculus Examples

$\ln ({(e)}^{2 + \ln ({(e)}^{3 \ln ({(e)}^{4 \ln ({(e)}^{5})})})})$

Step 1

Expand $\ln ({(e)}^{2 + \ln ({(e)}^{3 \ln ({(e)}^{4 \ln ({(e)}^{5})})})})$ by moving $2 + \ln ({(e)}^{3 \ln ({(e)}^{4 \ln ({(e)}^{5})})})$ outside the logarithm.

$(2 + \ln ({(e)}^{3 \ln ({(e)}^{4 \ln ({(e)}^{5})})})) \ln (e)$

Step 2

Simplify each term.

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Expand $\ln ({(e)}^{3 \ln ({(e)}^{4 \ln ({(e)}^{5})})})$ by moving $3 \ln ({(e)}^{4 \ln ({(e)}^{5})})$ outside the logarithm.

$(2 + 3 \ln ({(e)}^{4 \ln ({(e)}^{5})}) \ln (e)) \ln (e)$

Expand $\ln ({(e)}^{4 \ln ({(e)}^{5})})$ by moving $4 \ln ({(e)}^{5})$ outside the logarithm.

$(2 + 3 (4 \ln ({(e)}^{5}) \ln (e)) \ln (e)) \ln (e)$

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

$(2 + 3 (4 (5 \ln (e)) \ln (e)) \ln (e)) \ln (e)$

Multiply $\ln (e)$ by $\ln (e)$ by adding the exponents.

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Move $\ln (e)$ .

$(2 + 3 (4 (5 (\ln (e) \ln (e)))) \ln (e)) \ln (e)$

Multiply $\ln (e)$ by $\ln (e)$ .

$(2 + 3 (4 (5 \ln^{2} (e))) \ln (e)) \ln (e)$

Multiply $\ln^{2} (e)$ by $\ln (e)$ by adding the exponents.

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Move $\ln (e)$ .

$(2 + 3 (4 (5 (\ln (e) \ln^{2} (e))))) \ln (e)$

Multiply $\ln (e)$ by $\ln^{2} (e)$ .

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Raise $\ln (e)$ to the power of $1$ .

$(2 + 3 (4 (5 (\ln^{1} (e) \ln^{2} (e))))) \ln (e)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(2 + 3 (4 (5 {\ln (e)}^{1 + 2}))) \ln (e)$

Add $1$ and $2$ .

$(2 + 3 (4 (5 \ln^{3} (e)))) \ln (e)$

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

$(2 + 3 (4 (5 \cdot 1^{3}))) \ln (e)$

One to any power is one.

$(2 + 3 (4 (5 \cdot 1))) \ln (e)$

Multiply $3 (4 (5 \cdot 1))$ .

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Multiply $5$ by $1$ .

$(2 + 3 (4 \cdot 5)) \ln (e)$

Multiply $4$ by $5$ .

$(2 + 3 \cdot 20) \ln (e)$

Multiply $3$ by $20$ .

$(2 + 60) \ln (e)$

Step 3

Add $2$ and $60$ .

$62 \ln (e)$

Step 4

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

$62 \cdot 1$

Step 5

Multiply $62$ by $1$ .

$62$