Solving Logarithmic Equations: Analytical Method

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    Analytical Logarithms
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Discussion Overview

The discussion revolves around the analytical methods for solving logarithmic equations, specifically the equation x = (2^x)/14. Participants explore various approaches, including series expansions and special functions, while expressing skepticism about the existence of a true analytical solution.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about analytical methods for solving logarithmic equations, noting a preference for graphical approaches taught in high school.
  • Another suggests using the Taylor series expansion of 2^(x/14) for approximations, though this is met with some doubt by the original poster.
  • A participant claims there is no true analytical solution, mentioning the product log function as a way to express solutions but not as a genuine analytical method.
  • One reply discusses the use of approximation algorithms like the Newton-Raphson method, which is based on Taylor series expansions, and mentions the Lambert W function as a more advanced approach that is rarely used.
  • Humorous remarks are made regarding the Lambert W function and its historical context, adding a light-hearted tone to the discussion.
  • The original poster expresses a decision to continue using graphical methods due to a lack of access to calculators during tests.

Areas of Agreement / Disagreement

Participants express differing views on the existence and nature of analytical solutions for the logarithmic equation. While some suggest methods for approximation, others assert that a true analytical solution does not exist, indicating a lack of consensus.

Contextual Notes

The discussion includes references to various mathematical methods and functions, but the limitations of these approaches, such as the reliance on approximations and the absence of a definitive analytical solution, remain unresolved.

joo
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What is the analytical method for solving log. eqs., like x=2^x/14 ?

In high school they only teach us the graphical approach =/

joo
 
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What I meant was x=(2^x)/14, but I guess that doesn't really change the principle.

Thanks, I'll take a look at it, although I find myself doubtful.
 
There is no analytical solution to your problem. One can express the solutions using the product log function, but that is just another way of writing it, not a true analytic solution.
 
Welcome to PF, joo! :smile:In university they still use the graphical approach. ;)

In addition they use approximation algorithms, like the method of Newton-Raphson (which is based on a Taylor series expansion).

It's only the really bold ones in math that use the Lambert W function, which is a function that has only been invented to be able to write the solution to your equation.
As far as I know, no one really uses it.

The first solution for your equation is ##x=-{W(-\frac 1 {14} \ln(2)) \over \ln(2)} \approx 0.07525##.
 
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I like Serena said:
Welcome to PF, joo! :smile:


In university they still use the graphical approach. ;)

In addition they use approximation algorithms, like the method of Newton-Raphson (which is based on a Taylor series expansion).

It's only the really bold ones in math that use the Lambert W function, which is a function that has only been invented to be able to write the solution to your equation.
As far as I know, no one really uses it.

(HUMOR)

But Lambert used it and they made a movie on his life among sheep:



(/HUMOR)
 
Last edited by a moderator:
jedishrfu said:
(HUMOR)

But Lambert used it and they made a movie on his life among sheep:



(/HUMOR)


Oh! So the W comes from Walt Disney! :D
 
Last edited by a moderator:
Thank you for your replies ! I'll stick to the graphical solving for now then, since I will have no access to any calculators during my tests.
 

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