In deciding upon the appropriate premium to charge, insurance companies sometimes use the exponential principle, defined as follows. With X as the random amount that it will have to pay in claims, the premium charged by the insurance company is P = 1 a

ln(E[eaX])

where a is some specified positive constant. Find P when X is an exponential random variable with parameter λ, and a = αλ, where 0<α <1.