- #1
Little ant
- 21
- 0
Find the solutions of 5^x=2x+1 by ordinary methods? if it can´t be found by these ways, then prove that it's imposible.
Solve 5^x=2x+1 or Prove Impossibility
- Context: Undergrad
- Thread starter Little ant
- Start date
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Discussion Overview
The discussion revolves around finding solutions to the equation 5^x = 2x + 1, exploring both ordinary methods and the possibility of proving its impossibility. Participants examine various approaches, including graphical analysis and numerical methods.
Discussion Character
- Exploratory
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant suggests that by inspection, x = 0 is a solution.
- Another participant indicates that there is at least one other solution and questions how to find it.
- A different participant proposes studying the function y = (5^x) - (2x + 1) to identify roots, noting that one root is obvious.
- This participant also mentions that graphical representation can provide an approximation for the second root and suggests using numerical methods like Newton-Raphson for further accuracy.
- It is noted that analytical solutions may require the Lambert W function, which is outside the scope of ordinary methods.
Areas of Agreement / Disagreement
Participants generally agree that there are multiple solutions to the equation, with at least one being x = 0. However, the methods for finding additional solutions and the feasibility of proving impossibility remain contested.
Contextual Notes
Participants have not reached a consensus on the methods to be used or the existence of a definitive analytical solution. The discussion includes assumptions about the definition of "ordinary methods" and the applicability of numerical techniques.
- #2
stevenb
- 701
- 6
It depends on what you mean by ordinary methods.
By inspection, x=0.
By inspection, x=0.
- #3
Little ant
- 21
- 0
but, there is other answer, how you find it?
- #4
Femme_physics
Gold Member
- 2,548
- 1
The exponent ^5 is behind the X or after?
- #5
JJacquelin
- 801
- 35
Ordinary method consists in studying the function y=(5^x)-(2x+1)
This shows that two roots exist (one of them is obvious).
Drawing the graph of the function allows to obtain a first approximate of the second root.
Then, numerical computation leads to the value of the root, as accurate as we want. There are a lot of numerical methods : Newton-Raphson and many other...
Analytical solving is outside the scope of ordinary methods. It requires the use of a special function : the Lambert W function.
This shows that two roots exist (one of them is obvious).
Drawing the graph of the function allows to obtain a first approximate of the second root.
Then, numerical computation leads to the value of the root, as accurate as we want. There are a lot of numerical methods : Newton-Raphson and many other...
Analytical solving is outside the scope of ordinary methods. It requires the use of a special function : the Lambert W function.
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