Natural logarithm and pi help?

In summary, the conversation involves a discussion about proving the inequality ln(e)/e > ln(pi)/pi without calculating the values. The suggestion is made to consider the function f(x)=ln(x)/x with domain the positive real half-axis and determine its maximum value. It is then noted that ln(e)/e is the maximum value for f, and this can be used to prove the inequality e^pi > pi^e. The person seeking help expresses their understanding and feeling of foolishness.
  • #1
Guero
15
0
I haven't been able to prove:

ln(e)/e > ln(pi)/pi

without calculating any of the values. Help would be much appreciated.
 
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  • #2
Hint:
Consider the function
[tex]f(x)=\frac{ln(x)}{x}}[/tex],
with domain the positive real half-axis.

Determine the function's maximum value.
 
Last edited:
  • #3
mm, I can see that, but I was looking for a proof that shows that e^pi > pi^e
 
  • #4
Well, since you can prove that ln(e)/e is the maximum value for f, we also have:
[tex]\pi(ln(e))>eln(\pi)\to{ln}(e^{\pi})>ln(\pi^{e})[/tex]
wherefrom your inequality follows.
 
  • #5
Argh! I get it, Thanks!

I feel pretty stupid now.
 

Related to Natural logarithm and pi help?

1. What is a natural logarithm?

A natural logarithm is a mathematical function that is the inverse of the exponential function. It is denoted by "ln" and is used to calculate the power to which the base number, usually e, must be raised to obtain a given number.

2. How is natural logarithm related to e?

Euler's number, denoted by "e", is the base for natural logarithms. This means that the natural logarithm of a number is the power to which e must be raised to obtain that number.

3. What are the properties of natural logarithm?

The natural logarithm has several important properties, including:

  • The natural logarithm of 1 is 0.
  • The natural logarithm of e is 1.
  • The natural logarithm of a product is equal to the sum of the natural logarithms of the individual factors.
  • The natural logarithm of a quotient is equal to the difference of the natural logarithms of the individual terms.
  • The natural logarithm of a power is equal to the exponent times the natural logarithm of the base.

4. What is pi and how is it related to natural logarithm?

Pi, denoted by the Greek letter π, is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14. Pi is related to natural logarithm through the following formula: ln(x) = 2πi + ln|x|, where i is the imaginary unit and |x| is the absolute value of x.

5. How can natural logarithm and pi help in scientific calculations?

Natural logarithm and pi are important mathematical concepts used in many scientific calculations, such as in calculus, physics, and engineering. They are especially useful in exponential growth and decay equations, as well as in trigonometric functions.

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