Solve for x e^(2x)=25

$e^{2 x} = 25$

Step 1

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

$\ln (e^{2 x}) = \ln (25)$

Step 2

Expand the left side.

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

$2 x \ln (e) = \ln (25)$

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

$2 x \cdot 1 = \ln (25)$

Multiply $2$ by $1$ .

$2 x = \ln (25)$

Step 3

Divide each term in $2 x = \ln (25)$ by $2$ and simplify.

Divide each term in $2 x = \ln (25)$ by $2$ .

$\frac{2 x}{2} = \frac{\ln (25)}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\ln (25)}{2}$

Divide $x$ by $1$ .

$x = \frac{\ln (25)}{2}$

Simplify the right side.

Rewrite $\ln (25)$ as $\ln (5^{2})$ .

$x = \frac{\ln (5^{2})}{2}$

Expand $\ln (5^{2})$ by moving $2$ outside the logarithm.

$x = \frac{2 \ln (5)}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \frac{2 \ln (5)}{2}$

Divide $\ln (5)$ by $1$ .

$x = \ln (5)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \ln (5)$

Decimal Form:

$x = 1.60943791 \dots$