How to prove ln(x) < sqrt(x) for all x>0

Hi,

I know I can use a graphical calculator to easily show that
How to prove ln(x) < sqrt(x) for all x>0

But I wonder if there is a rigorous way to demonstrate this.
 
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what do you call that? proof by calculator graphic? I would consider the function [itex]f(x) = lnx - \sqrt{x}[/itex] (or is it the other way around? well you get the idea) & use calculus to show that the resulting function is monotone increasing on that interval. off the top of my head that's probably the best way to do it.
 
Is there a non-graphical way of doing it?
 
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Hi Grossgermany,

To answer the question, I'm assuming you have a rough idea of Calculus.

We know that the derivative of a formula, tells us the rate of change of the formula. We also that both formulas are always positive.

(Sorry no Latex I tried to make it works, but i failed. )

So let's take the derivative of ln(x) d/dx (ln(x)) = 1/x
Then of sqrt(x) d/dx (sqrt(x)) = 1/(2sqrt(x))

As x goes to infinity, 1/2 becomes useless, and for simplicity sake let's remove it.

1/x < 1/(2sqrt(x)) --> 1/x < 1/sqrt(x)

Then we can clearly see that as x increase to infinity, the rate of change of sqrt(x) is greater then ln(x).

I've just though about it while reading your question, but I am fairly sure of my answer.

Vardd
 
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Showing the derivative of x^1/2 – ln(x) is positive for all x>0 is fairly easy.
 
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Hi Grossgermany,

To answer the question, I'm assuming you have a rough idea of Calculus.

We know that the derivative of a formula, tells us the rate of change of the formula. We also that both formulas are always positive.

(Sorry no Latex I tried to make it works, but i failed. )

So let's take the derivative of ln(x) d/dx (ln(x)) = 1/x
Then of sqrt(x) d/dx (sqrt(x)) = 1/(2sqrt(x))

As x goes to infinity, 1/2 becomes useless, and for simplicity sake let's remove it.

1/x < 1/(2sqrt(x)) --> 1/x < 1/sqrt(x)

Then we can clearly see that as x increase to infinity, the rate of change of sqrt(x) is greater then ln(x).

I've just though about it while reading your question, but I am fairly sure of my answer.

Vardd
You can’t just remove the ½, and your method would only show for some value of N, for x>N would ln(x) < x^1/2
 
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Showing the derivative of x^1/2 – ln(x) is positive for all x>0 is fairly easy.
This is the way to do it. You also need to show that it's a positive value from the beginning though (lim x approaches 0 of f(x) = +) since increasing wouldn't really prove anything by itself (ex: any function where f(x) = - where x< some positive number k, but f(x) increasing from x>0).
 
This is the way to do it. You also need to show that it's a positive value from the beginning though (lim x approaches 0 of f(x) = +) since increasing wouldn't really prove anything by itself (ex: any function where f(x) = - where x< some positive number k, but f(x) increasing from x>0).
Except the statement is not true
0.5*1/sqrt(x)-1/x is not always positive for x>0.
 
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whoops, it's not monotone like I said but the minimum is positive, & that will still do it
 
consider the function [tex]{f(x) = e^{\sqrt{x}} - x,}[/tex] which is >0
so that [tex]f^{\prime}[/tex] >0 for all x and therefore
[tex]e^{\sqrt{x}} - x > 0[/tex] , and thus [tex]\sqrt{x} > \ln{x}[/tex]
 

CRGreathouse

Science Advisor
Homework Helper
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whoops, it's not monotone like I said but the minimum is positive, & that will still do it
consider the function [tex]{f(x) = e^{\sqrt{x}} - x,}[/tex] which is >0
so that [tex]f^{\prime}[/tex] >0 for all x and therefore
[tex]e^{\sqrt{x}} - x > 0[/tex] , and thus [tex]\sqrt{x} > \ln{x}[/tex]
How do you prove that it's positive?
 

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