- #1
physicsss
- 319
- 0
I've been trying to do this for almost an hour now but I'm still not able to show that [1-e^(-x)]/[1-e^(x)]=-e^(-x).
Thank you.
Thank you.
Proving the Equation: [1-e^(-x)]/[1-e^(x)]=-e^(-x)
- Context: Undergrad
- Thread starter physicsss
- Start date
Click For Summary
Discussion Overview
The discussion revolves around proving the equation \(\frac{1-e^{-x}}{1-e^{x}}=-e^{-x}\). Participants explore various algebraic manipulations and approaches to demonstrate the equivalence of their derived expression and the one provided in a textbook.
Discussion Character
- Mathematical reasoning
- Debate/contested
- Homework-related
Main Points Raised
- One participant expresses difficulty in proving the equation and seeks assistance.
- Another suggests cross-multiplying as a potential method to simplify the expression.
- A participant clarifies that they arrived at the expression \(\frac{1-e^{-x}}{1-e^{x}}\) but believes it is equivalent to \(-e^{-x}\) as stated in the textbook.
- One participant proposes multiplying both the numerator and denominator by \(e^{x}\) to facilitate the proof.
- Another participant claims that their method works, showing the steps leading to \(-e^{-x}\) through algebraic manipulation.
- A participant questions the validity of a step taken by another, specifically regarding the transformation of \((e^{x}-1)/(1-e^{x})\).
- Further clarification is provided regarding the transformation of the expression, with a participant asserting they turned it into \(-1\) rather than \(1\).
- Another participant suggests multiplying the original equation by \([1-e^{x}]\) and provides a series of steps that lead back to the original expression.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the best method to prove the equation, with multiple approaches and some disagreement on the validity of specific steps in the algebraic manipulations.
Contextual Notes
Some participants express uncertainty about the correctness of their algebraic steps, and there are unresolved questions regarding the transformations used in the proofs.
- #2
dextercioby
Science Advisor
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U mean show that
[tex]\frac{1-e^{-x}}{1-e^{x}}=-e^{-x}[/tex]
How about cross multiply??
Daniel.
[tex]\frac{1-e^{-x}}{1-e^{x}}=-e^{-x}[/tex]
How about cross multiply??
Daniel.
- #3
physicsss
- 319
- 0
No, no. I got [1-e^(-x)]/[1-e^(x)] as an answer to a problem, but in the back of the book the answer is given as -e^(-x). What, I'm asking then, is how to get from my answe to the book's...I know for a fact that they're equal.
Sorry for the confusion.
Sorry for the confusion.
- #4
dextercioby
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How about multiplying both the numerator and the denominator by [itex]e^{x}[/itex] ??
Daniel.
Daniel.
- #5
physicsss
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It doesn't work. 
- #6
dextercioby
Science Advisor
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It works for me
[tex](\frac{1-e^{-x}}{1-e^{x}})(\frac{e^{x}}{e^{x}})=(\frac{e^{x}-1}{1-e^{x}})(\frac{1}{e^{x}})=-1\cdot \frac{1}{e^{x}}=-e^{-x}[/tex]
Daniel.
[tex](\frac{1-e^{-x}}{1-e^{x}})(\frac{e^{x}}{e^{x}})=(\frac{e^{x}-1}{1-e^{x}})(\frac{1}{e^{x}})=-1\cdot \frac{1}{e^{x}}=-e^{-x}[/tex]
Daniel.
- #7
Eratosthenes
- 73
- 0
(1 - e^(-x))/(1 - e^x) = (1 - 1/e^x)/(1 - e^x) ...then find a common denominator on the top fraction and you get [(e^x - 1)/e^x]/(1 - e^x) = (e^x - 1)/[(e^x)(1 - e^x)] then -1*(1-e^x)/[(e^x)(1 - e^x)] = -1/e^x = -e^(-x)
wow i typed that fast, should be right,i included lots of steps, i skipped a few, but included more than what you'd need to write down probably, do you see it now?
wow i typed that fast, should be right,i included lots of steps, i skipped a few, but included more than what you'd need to write down probably, do you see it now?
- #8
physicsss
- 319
- 0
@dextercioby:
how did you turn (e^x-1)/(1-e^x) into 1? If you multiply it by -1 on the top don't you have to do it to the bottom too?
how did you turn (e^x-1)/(1-e^x) into 1? If you multiply it by -1 on the top don't you have to do it to the bottom too?
- #9
dextercioby
Science Advisor
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physicsss said:
I didn't...I turned into "-1"...
Daniel.
- #10
Edgardo
- 707
- 17
Hello Physicsss,
multiply your equation [1-e^(-x)]/[1-e^(x)]=-e^(-x)
by [1-e^(x)].
You get
[1-e^(-x)] = -e^(-x) * [1-e^(x)]
=> [1-e^(-x)] = -e^(-x) + (-e^(-x))*(-e^(x))
=>[1-e^(-x)] = -e^(-x) + 1
=> [1-e^(-x)] = 1-e^(-x)
multiply your equation [1-e^(-x)]/[1-e^(x)]=-e^(-x)
by [1-e^(x)].
You get
[1-e^(-x)] = -e^(-x) * [1-e^(x)]
=> [1-e^(-x)] = -e^(-x) + (-e^(-x))*(-e^(x))
=>[1-e^(-x)] = -e^(-x) + 1
=> [1-e^(-x)] = 1-e^(-x)
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