Precalculus Examples

$\ln (\frac{3 e}{2 \sqrt{x}})$

Step 1

Rewrite $\ln (\frac{3 e}{2 \sqrt{x}})$ as $\ln (3 e) - \ln (2 \sqrt{x})$ .

$\ln (3 e) - \ln (2 \sqrt{x})$

Step 2

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

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

Step 3

Rewrite $\ln (3 e)$ as $\ln (3) + \ln (e)$ .

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

Step 4

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

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

Step 5

Rewrite $\ln (2 x^{\frac{1}{2}})$ as $\ln (2) + \ln (x^{\frac{1}{2}})$ .

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

Step 6

Expand $\ln (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\ln (3) + 1 - (\ln (2) + \frac{1}{2} \ln (x))$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{1}{2}$ and $\ln (x)$ .

$\ln (3) + 1 - (\ln (2) + \frac{\ln (x)}{2})$

Step 7.2

Apply the distributive property.

$\ln (3) + 1 - \ln (2) - \frac{\ln (x)}{2}$