Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.

a. ln xy = ln x + ln y. 

b. ln 0 = 1.

c. ln (x + y) = ln x + ln y. 

d. 2^x = e^{2 ln x}.
e. The area under the curve y = 1/x and the x-axis on the interval [1, e] is 1.