Precalculus Examples

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

Step 1

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

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

Step 2

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 3

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$4.34326863 \dots$