Solve for x e^(0.4x)=0.4

$e^{0.4 x} = 0.4$

Step 1

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

$\ln (e^{0.4 x}) = \ln (0.4)$

Step 2

Expand the left side.

Expand $\ln (e^{0.4 x})$ by moving $0.4 x$ outside the logarithm.

$0.4 x \ln (e) = \ln (0.4)$

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

$0.4 x \cdot 1 = \ln (0.4)$

Multiply $0.4$ by $1$ .

$0.4 x = \ln (0.4)$

Step 3

Divide each term in $0.4 x = \ln (0.4)$ by $0.4$ and simplify.

Divide each term in $0.4 x = \ln (0.4)$ by $0.4$ .

$\frac{0.4 x}{0.4} = \frac{\ln (0.4)}{0.4}$

Simplify the left side.

Cancel the common factor of $0.4$ .

Cancel the common factor.

$\frac{0.4 x}{0.4} = \frac{\ln (0.4)}{0.4}$

Divide $x$ by $1$ .

$x = \frac{\ln (0.4)}{0.4}$

Simplify the right side.

Replace $e$ with an approximation.

$x = \frac{\log_{2.71828182} (0.4)}{0.4}$

Log base $2.71828182$ of $0.4$ is approximately $- 0.91629073$ .

$x = \frac{- 0.91629073}{0.4}$

Divide $- 0.91629073$ by $0.4$ .

$x = - 2.29072682$