Proving e^x=x^e for only one positive x

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In summary, the problem is to prove that the equation x^e=e^x has only one positive solution. The relevant equations include the base change for logarithms. The attempt at a solution involved using the properties of equal bases and assuming x=e+a, but this did not lead to a clear solution. Instead, the suggestion is to use inequalities and graph the functions x^(1/x) and e^(1/e), which will show that they intersect only at x=e. Raising both functions to the power of x*e will prove that x=e is the only positive solution.
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desquee
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Hi, I'm teaching myself calc 2, and could use some help with a problem I'm not sure how to solve:

Problem:
Prove that x^e=e^x has only one positive solution.

Relevant equations (I think?):

b^x = a^(loga(b)*x) - base change for logarithms

The attempt at a solution:

e^x=x^e=e^(ln(x)*e) - base change
x = ln(x)*e - powers of equal bases
At this point I'm stuck, I'm not sure how to show that x = ln(x)*e has only one positive solution.
 
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  • #2
Try to argue for what happens when x < e and when x > e, respectively. (Obviously, x = e is a solution.)
 
  • #3
I tried assuming x=e+a, and get e^e+a=(e+a)e=eln(e+a)*e, which gives (e+a)/e=ln(e+a), or e(e+a)/e=e+a, but I haven't been able to reach a clear absurdity.
 
  • #4
Try to use inequalities rather than working with a constant.
 
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  • #5
Try plotting graph of x∧(1/x) for x>0 .

You will see that it is increasing for x<e and decreasing for x>e .

Now compare x∧(1/x) with e∧(1/e) . The same value is only possible for x=e .

Raise both to x*e and you will get your answer .
 
  • #6
I always start these deals with graphing.

Once you graph it and see that the functions are increasing at different rates, it will become clear how to prove they only intersect once.
 

Related to Proving e^x=x^e for only one positive x

What does "Proving e^x=x^e for only one positive x" mean?

"Proving e^x=x^e for only one positive x" is a mathematical statement that asks whether there is a real number, specifically a positive number, that can be raised to a power (x) and have the same value as the base (e) raised to the power of that number (e^x).

Why is this statement significant?

This statement is significant because it relates to the properties of exponential and logarithmic functions, which have many practical applications in fields such as finance, physics, and biology. It also has connections to the famous Euler's Identity, which states that e^(iπ) = -1.

Is it possible to prove this statement for more than one positive x?

No, it is not possible to prove this statement for more than one positive x. This is because the function e^x=x^e only has one solution for positive x, which is x=e. This can be proven using calculus and the properties of logarithms.

How is this statement typically proved?

This statement is typically proved using calculus, specifically by taking the derivative of both sides of the equation and showing that they are equal. This is known as the logarithmic differentiation method. It can also be proved using algebraic manipulation and the properties of logarithms.

What are the practical applications of this statement?

The practical applications of this statement are mainly in the fields of mathematics and physics. It is also used in finance for calculating compound interest and in biology for modeling population growth. Additionally, this statement has connections to the Gaussian integral and the normal distribution, which are important in statistics and probability theory.

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