Precalculus Examples

$\ln (x \cdot e^{4 x}) (x)$

Step 1

Rewrite $\ln (x \cdot e^{4 x})$ as $\ln (x) + \ln (e^{4 x})$ .

$(\ln (x) + \ln (e^{4 x})) x$

Step 2

Use logarithm rules to move $4 x$ out of the exponent.

$(\ln (x) + 4 x \ln (e)) x$

Step 3

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

$(\ln (x) + 4 x \cdot 1) x$

Step 4

Multiply $4$ by $1$ .

$(\ln (x) + 4 x) x$

Step 5

Apply the distributive property.

$\ln (x) x + 4 x \cdot x$

Step 6

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1

Move $x$ .

$\ln (x) x + 4 (x \cdot x)$

Step 6.2

Multiply $x$ by $x$ .

$\ln (x) x + 4 x^{2}$

Step 7

Reorder factors in $\ln (x) x + 4 x^{2}$ .

$x \ln (x) + 4 x^{2}$

Step 8

Reorder $x \ln (x)$ and $4 x^{2}$ .

$4 x^{2} + x \ln (x)$