Using jacobian method finding the probability density function

Let X ~ Exponential (λ), for some constant λ > 0. That is,
fX(x) = λ e-λx = λ exp( -λx ), x > 0, ( fX(x) = 0 otherwise)

(a) Create a transformed random variable Y = ln(X) (natural log of X). Using either the CDF method or Jacobian method, show that the probability density function of Y is given by:
fY(y) = λ exp( y - λey ), -∞ ≤ y ≤ ∞.

(b) Verify that the CDF of Y is FY(y) = 1 - exp (-λey ), -∞ ≤ y ≤ ∞.

(c) If λ = 1, compute the median of Y.