Precalculus Examples

$\ln (\frac{x}{e^{x}})$

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

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

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

$\ln (x e^{x \cdot - 1})$

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

$\ln (x) + \ln (e^{x \cdot - 1})$

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

$\ln (x) + x \cdot - 1 \ln (e)$

Move $- 1$ to the left of $x$ .

$\ln (x) - 1 \cdot x \ln (e)$

Rewrite $- 1 x$ as $- x$ .

$\ln (x) - x \ln (e)$

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

$\ln (x) - x \cdot 1$

Multiply $- 1$ by $1$ .

$\ln (x) - x$

Reorder $\ln (x)$ and $- x$ .

$- x + \ln (x)$