Prove ln(x) < sqrt(x) for all x>0

[grossgermany]

Hi,

I wonder how to prove that ln(x) < sqrt(x) for all x>0?

Please enlighten me on two possible way to prove this .
Proof1. Using calculus and derivatives
Proof2. Since I'm taking real analysis, I wonder if it is possible to use taylor series to show this in an elegant way.

 

[Petr Mugver]

I don't know a "proof2", here's a "proof1" (idea, you do the calculations!):

1) Prooving your statement is equivalent to prooving that f(x) = ln(x) - sqrt(x) < 0 for all x > 0.

2) Calculate f ' (x).

3) Find y such that f ' (y) = 0.

4) Show that f (y) < 0.

5) Show that the limits for x -> 0+ and x -> +infty of f(x) are negative.

6) Why points 1) - 5) proove your claim?

 

[Hurkyl]

You already have a thread on this.