Solve for x e^(2x)=6

$e^{2 x} = 6$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 x}) = \ln (6)$

Step 2

Expand the left side.

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

$2 x \ln (e) = \ln (6)$

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

$2 x \cdot 1 = \ln (6)$

Multiply $2$ by $1$ .

$2 x = \ln (6)$

Step 3

Divide each term in $2 x = \ln (6)$ by $2$ and simplify.

Divide each term in $2 x = \ln (6)$ by $2$ .

$\frac{2 x}{2} = \frac{\ln (6)}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\ln (6)}{2}$

Divide $x$ by $1$ .

$x = \frac{\ln (6)}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (6)}{2}$

Decimal Form:

$x = 0.89587973 \dots$