Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \ln (\sqrt{e})$ by moving $2$ inside the logarithm.

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

Step 1.2

Rewrite ${\sqrt{e}}^{2}$ as $e$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{e}$ as $e^{\frac{1}{2}}$ .

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

Step 1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.2.5

Simplify.

$\ln (e) + e^{\ln (3)}$

Step 1.3

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

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

Step 1.4

Exponentiation and log are inverse functions.

$1 + 3$

Step 2

Add $1$ and $3$ .

$4$