Relationship between density function

Question

Could someone help me on this question. I'm not entirely sure how to relate the two different density functions to each other.

Answer 1

First let's find $F_Y(y)$ with the definition

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

Now let's calculate its derivative to get the requested density

$$f_Y(y)=f_X(log(y))\cdot \frac{1}{y}\mathbb{1}_{(e;e^2)}(y)$$

...that is (a)