Inequality: If $\alpha>1$, is $\log^\alpha(x)\leq x$ as $x \rightarrow \infty$?

Question

Let $\alpha>1$. Is $\log^\alpha(x)\leq x$ when $x \rightarrow \infty$?

Answer 1

Hint:

• the logarithm is an increasing function

• so consider $y=\log(x)$ and

• whether $y^\alpha\leq e^y$ when $y \rightarrow \infty$ ?

Answer 2

Yes. From the power series, $e^x > x^n/n!$.

Take logs and choose $n > \alpha+1$.