Solve for x e^(0.5x)=0.5

$e^{0.5 x} = 0.5$

Step 1

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

$\ln (e^{0.5 x}) = \ln (0.5)$

Step 2

Expand the left side.

Expand $\ln (e^{0.5 x})$ by moving $0.5 x$ outside the logarithm.

$0.5 x \ln (e) = \ln (0.5)$

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

$0.5 x \cdot 1 = \ln (0.5)$

Multiply $0.5$ by $1$ .

$0.5 x = \ln (0.5)$

Step 3

Divide each term in $0.5 x = \ln (0.5)$ by $0.5$ and simplify.

Divide each term in $0.5 x = \ln (0.5)$ by $0.5$ .

$\frac{0.5 x}{0.5} = \frac{\ln (0.5)}{0.5}$

Simplify the left side.

Cancel the common factor of $0.5$ .

Cancel the common factor.

$\frac{0.5 x}{0.5} = \frac{\ln (0.5)}{0.5}$

Divide $x$ by $1$ .

$x = \frac{\ln (0.5)}{0.5}$

Simplify the right side.

Replace $e$ with an approximation.

$x = \frac{\log_{2.71828182} (0.5)}{0.5}$

Log base $2.71828182$ of $0.5$ is approximately $- 0.69314718$ .

$x = \frac{- 0.69314718}{0.5}$

Divide $- 0.69314718$ by $0.5$ .

$x = - 1.38629436$