Proof of a constant

Question

Assignment:

Let α > 0. Prove that log x ≤ x^α for x large. Prove that there exists a constant C_α such that log x ≤ C_αx^α for all x ∈ [1,∞), C_α → ∞ as α → 0+, and C_α → 0 as α → ∞.

Please justify all steps and be rigorous because it is an analysis problem. (Note: The problem falls under the chapter on Differentiability on R in the section entitled The Mean Value Theorem, and the hint says: Find the maximum of f(x) = log x/x^αforx ∈ [1,∞))

Provide complete and step by step solution for the question and show calculations and use formulas.