Density function of product of random variables

[khotsofalang]

suppose you have two random variables X and Y which are independent,
we want to form a new random variable Z=XY, if f(x) and f(y) are density functions
of X and Y respectively what is the density function of Z?

I tried taking logs and applying convolution, but it did not really work

 

[statdad]

It isn't clear to me from your writing whether X and Y have the same distribution - you used the same name for their densities - but think about this start: it will work whether they are identically distributed or not. Start with this
$$P(Z \le z) = E_Y \left[P\left( XY \le z \mid Y = y\right) \right]$$

and see what develops.