Let M be a large real number. Explain briefly why there must be exactly one root A of the equation Mx = e^x with A > 1. Why is logM a reasonable approximation to A? Write

A = logM + y. Can you give an approximation to y, and hence improve on logM as an approximation to A?

Im not sure whether what I've done is correct:

The graph of y=Mx and y=e^x only intersects once, as e^x is a much greater increasing function.

M is a large number, so M=e^x approximately, so lnM = x approximately

A = logM + y

so y=logA

Thanks