- #1

desquee

- 18

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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:

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

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

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.

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

Relevant equations (I think?):Relevant equations (I think?):

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

The attempt at a solution: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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