Full solution

\frac{1}{2} = e^{-\frac{2}{x}}

Identify Equation & Goal: Identify the equation and the goal. $\newline$ We are given the equation $\frac{1}{2} = e^{-\frac{2}{x}}$ , and we need to solve for $x$ .
Take Natural Logarithm: Take the natural logarithm of both sides to eliminate the exponential function. $\newline$ $\ln(\frac{1}{2}) = \ln(e^{-\frac{2}{x}})$
Use Logarithm Property: Use the property of logarithms that $\ln(e^y) = y$ . $\ln(\frac{1}{2}) = -\frac{2}{x}$
Solve for x: Solve for x by multiplying both sides by $-x$ and dividing by $\ln(\frac{1}{2})$ . $\newline$ $x = \frac{-2}{\ln(\frac{1}{2})}$
Calculate $\ln(\frac{1}{2})$ : Calculate the value of $\ln(\frac{1}{2})$ . $\newline$ $\ln(\frac{1}{2})$ is the natural logarithm of $\frac{1}{2}$ , which is a negative value because $\frac{1}{2}$ is less than $1$ . $\newline$ $\ln(\frac{1}{2}) \approx -0.693147$
Substitute $\ln(\frac{1}{2})$ into $x$ : Substitute the value of $\ln(\frac{1}{2})$ into the equation for $x$ . $x = \frac{-2}{(-0.693147)}$
Calculate $x$ : Calculate the value of $x$ . $x \approx \frac{-2}{(-0.693147)} \approx 2.88539$