Precalculus Examples

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

Step 1

Rewrite $\frac{\sqrt{2} x^{\frac{3}{2}}}{e^{2}}$ as $\sqrt{2} x^{\frac{3}{2}} {(e^{2})}^{- 1}$ .

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

Step 2

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

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

Step 3

Rewrite $\ln (\sqrt{2} x^{\frac{3}{2}} e^{2 \cdot - 1})$ as $\ln (\sqrt{2} x^{\frac{3}{2}}) + \ln (e^{2 \cdot - 1})$ .

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

Step 4

Use logarithm rules to move $2 \cdot - 1$ out of the exponent.

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

Step 5

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

Multiply $- 2$ by $1$ .

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