Math Help - More Proofs

1. More Proofs

Let $\alpha > 0$. Prove that no matter how small $\alpha$ is, there is an $N \epsilon$ R such that $ln x \leq x^{\alpha}$ for all $x \leq N$. Note: N depends on $\alpha$.

I need to use the mean value theorem to do this.
I started by finding the difference of the derivatives of both functions.
Then didnt know where to go... Any help would be greatly appreciated...

2. Fixed Latex error... sorry about that

3. I've been playing around with this problem... This is what I have so far...

$g' (x) - f' (x)$ is decreasing when $\alpha$ is very small.

For $\alpha = \frac{1}{200}$ or smaller $x = 2.27$

The inequality only holds true for $x\leq 2.27$and I need to find a $N\leq x$

Can anyone help me with explaining this or let me know if I am doing something wrong please??