Solving Logarithmic Equations: Analytical Method

[joo]

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

 

[jedishrfu]

I suppose you could replace the 2^(x/14) with its taylor series expansion and then cut-off some terms to get an approximation:

http://en.wikipedia.org/wiki/Taylor_series

 

[joo]

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.

 

[0xDEADBEEF]

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.

 

[I like Serena]

Welcome to PF, joo! 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 $x=-{W(-\frac 1 {14} \ln(2)) \over \ln(2)} \approx 0.07525$.

 

[jedishrfu]

Welcome to PF, joo!


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)

 

[I like Serena]

(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

 

[joo]

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.