Equation in "e" difficult to solve.

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luismorana

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Hello, first time here.
Here is the equation I want to understand how to solve without Calculus or using Mat-lab, etc. I know that using math software it is easy and fast but I need to understand how to analytically solve it. I tried all I knew about "ln" and cannot get to the final solution. Sometime ago, others helped me with a solution involving Calculus, which it was not easy for me to understand. If there is a way to find the solution without calculus, it will be greatly appreciated.

0.5= (1-e^(-0.9/x)) / (1-e^(-1/x))

I need to understand how to calculate "x". In this case I know, from the book I am reading, that x= -0.1444 and if you replace this value in the equation, the final value is 0.5.

I would like to write a small piece of Software to calculate "x".

Thanks in advance
Luis
 
luismorana said:
If there is a way to find the solution without calculus, it will be greatly appreciated.

0.5= (1-e^(-0.9/x)) / (1-e^(-1/x))

I would like to write a small piece of Software to calculate "x".
Click to expand...

Hi Luis.

Taking the xth root of powers of e transcends algebra. That is, we cannot solve this equation algebraically.

If there is another method beyond calculus, I don't know what it is. You desire to write a program to estimate the Real value of x to some number of decimal places (and ignore all of the other Complex solutions), yes?

If you know how to code for the zeros of 9th-degree polynomials, perhaps the following will help you to estimate x.

This is a solution from other software (MVR5):

\(\displaystyle x = -\dfrac{1}{10} \cdot \dfrac{1}{ln(ZerosOf(Z^9 - Z^8 - Z^7 - Z^6 - Z^5 - Z^4 - Z^3 - Z^2 -Z - 1))}\)

Cheers :cool:
 
Thank you Gentlemen,
You gave me a good hint to move on the programming side of the solution and also some understanding on how to find out the solution. I will work on a routine according to the guidelines from you.

Regarding how to solve the equation algebraically, mmm4444bot is correct since I believe there is no way to find an analytical (or direct)
method to involve the use of closed-form equations to obtain an exact solution.
The obvious way to approach this it is numerical methods as suggested by the author of the book.

Thank you for your time and consideration.
Luis.:D
 
Denis said:
Smoke comes out!
Click to expand...

Clear smoke, I hope.

Now that that's cleared up, my lowest value for d after the first loop is +1. Did you expect that?

Just to be clear, here's my d:

\(\displaystyle -1-{e^{-{x}^{-1}}}+2\,{e^{- 0.9\,{x}^{-1}}}\)

To clear this up, perhaps you meant to type "highest value of d" (that's at x = -1)? or maybe d=b-a?

Clearly, I don't get it winker.gif
 
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