Precalculus Examples

$\ln ({(\frac{x}{e})}^{6 x})$

Step 1

Apply the product rule to $\frac{x}{e}$ .

$\ln (\frac{x^{6 x}}{e^{6 x}})$

Step 2

Rewrite $\frac{x^{6 x}}{e^{6 x}}$ as $x^{6 x} {(e^{6 x})}^{- 1}$ .

$\ln (x^{6 x} {(e^{6 x})}^{- 1})$

Step 3

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

$\ln (x^{6 x} e^{6 x \cdot - 1})$

Step 4

Rewrite $\ln (x^{6 x} e^{6 x \cdot - 1})$ as $\ln (x^{6 x}) + \ln (e^{6 x \cdot - 1})$ .

$\ln (x^{6 x}) + \ln (e^{6 x \cdot - 1})$

Step 5

Use logarithm rules to move $6 x \cdot - 1$ out of the exponent.

$\ln (x^{6 x}) + 6 x \cdot - 1 \ln (e)$

Step 6

Multiply $- 1$ by $6$ .

$\ln (x^{6 x}) - 6 x \ln (e)$

Step 7

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

$\ln (x^{6 x}) - 6 x \cdot 1$

Step 8

Multiply $- 6$ by $1$ .

$\ln (x^{6 x}) - 6 x$

Step 9

Reorder $\ln (x^{6 x})$ and $- 6 x$ .

$- 6 x + \ln (x^{6 x})$