If $X$ Has CDF F, What Is The CDF Of $Y=e^X$?

Question

If $X$ Has CDF F, What Is The CDF of $Y=e^X$?

Frankly, I'm not even sure where to start. Any pointers would be very helpful.

Answer 1

I'm not even sure where to start

If I was in you I would start with the definition of CDF

$$F_Y(y)=\mathbb{P}(Y\leq y)=\mathbb{P}(e^X\leq y)=\mathbb{P}(X\leq \log y)=F_X(\log y)$$

Answer 2

Start with the definition. The CDF of $e^X$ is the function $F_{e^X}(t) = \mathbb P(e^X \le t)$. So what does $e^X \le t$ tell you about $X$?