I got its integral to get the CDF, which is equal to $-e^{-x}$, and I equate it to 0.5, but I'm stuck since I was getting an answer of ln(-0.5) which is a math error.
2026-03-26 10:45:14.1774521914
What is the median of the pdf $f(x) = e ^{-x}$?
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The problem with your reasoning is that the integral under the curve $e^{-x}$ from $-\infty$ to $\infty$ is actually infinite. However, if you limit the domain of your pdf to only from $0$ to $\infty$ the integral over the entire pdf would be $1$ as desired.
So if you set the equation to be integrating from $0$ to $a$ of $e^{-x}$ to be equal to $0.5$, then you would find the correct value of the median for this pdf!
$$\int_0^a{e^{-x}}=\frac12$$