Let a > 0 and recall that (x) = ax-1 and (log x)' = 1/x for all

Question:

Let a > 0 and recall that (xα)ʹ = axα-1 and (log x)' = 1/x for all x > 0.
a) Prove that log x < xa for x large. Prove that there exists a constant Ca such that log x < Caxa for all x ∈ [1, ∞), Ca → ∞ as α → 0+, and Ca → 0 as a → ∞.
b) Obtain an analogue of part a) valid for ex and xa in place of log x and xα.