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

• grossgermany
Please do not post the same question multiple times. In summary, the conversation discusses two possible ways to prove that ln(x) < sqrt(x) for all x>0: using calculus and derivatives, or using Taylor series. The first method involves showing that the function f(x) = ln(x) - sqrt(x) is negative for all x>0, while the second method is still being explored.
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.

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?

## 1. What is the statement being proved?

The statement being proved is that the natural logarithm of x is always less than the square root of x for all positive values of x.

## 2. Why is this statement important?

This statement is important because it is a fundamental inequality in mathematics and has many applications in fields such as calculus, probability, and number theory.

## 3. What is the proof for this statement?

The proof for this statement involves using basic properties of logarithms and inequalities, such as the fact that ln(x) = 2ln(sqrt(x)) and the fact that the natural logarithm function is strictly increasing. By manipulating these properties, we can show that ln(x) < sqrt(x) for all positive values of x.

## 4. Are there any exceptions to this statement?

No, there are no exceptions to this statement. The inequality holds true for all positive values of x, regardless of their magnitude.

## 5. How can this statement be applied in real-world situations?

This statement can be applied in many real-world situations, such as calculating interest rates, analyzing data in statistics, and solving differential equations in physics and engineering. It is also used in the proof of the Central Limit Theorem in probability theory.

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