Solve for x e^(5x)=2

$e^{5 x} = 2$

Step 1

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

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

Step 2

Expand the left side.

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

$5 x \ln (e) = \ln (2)$

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

$5 x \cdot 1 = \ln (2)$

Multiply $5$ by $1$ .

$5 x = \ln (2)$

Step 3

Divide each term in $5 x = \ln (2)$ by $5$ and simplify.

Divide each term in $5 x = \ln (2)$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.13862943 \dots$