Re: How to solve this natural log problem?


"David W. Cantrell" <DWCantrell@xxxxxxxxxxx> wrote in message news:20070129122419.199$HL@xxxxxxxxxxxxxxxxx
"Dirk Van de moortel" <dirkvandemoortel@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
<amy_burton2007@xxxxxxxxx> wrote in message
news:1170055954.369782.189430@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> How to solve for x?
>
> 4000x = -6.9 - ln x

Take excel and write a number in cell A1.
In cell A2 enter the formula
= A1 - ( 4000*A1 + 6.9 + LN(A1) ) / ( 4000 + 1/A1 )
and copy this down to about A20.

This is Newton's method with f(x) = 4000 x + 6.9 + ln(x)

You're going to need a very small number to start with in A1
Try putting 1, 0.1, 0.01, 0.001 etc... in A1 and see what happens.

If you don't have excel, just take a high precision calculator
and you'll find the result in a few minutes.

My first preference would be to use the Lambert W function, as Rob did.

Yes, although excel doesn't have the Lambert W function (and
I have *never* seen a home/garden/kitchen calculator offer it),
there is, at least on this group, a strange tendency to mention
Lambert W whenever someone is looking for a solution to this
kind of equations.
Strange :-)



My second preference would be to use some analysis to get an approximate
solution. Then, if that's not sufficiently accurate, it at least gives a
good value for A1 in Newton's (or some other iterative) method.

Knowing that x must be a small positive number, the equation

a x = -b - ln(x)

has the approximate solution

x = (2a + 3c - Sqrt(-2a^2 + 24ac + 9c^2)) / (a(a - 2c)) where c = e^b.

Using a = 4000 and b = 6.9 in the above, we get the approximate solution

x = 0.000308... For comparison, the solution, as mentioned by Rob, is

x = 0.0003016...

Yes, that's what Excel gives me after a mere 5 iterations
with a seed of 0.001.

Dirk Vdm
.

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