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Help to figure out equation solution

by sj21
Tags: equation, figure, solution
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Mar15-12, 12:08 AM
P: 6

While working on a project I came across an equation and need some help to solve it.
This is the equation

T - log(T) = 1-R

Where, T = variable whose value is to be found
log base is 10
R = given value so basically right side of equal to sign is a constant..

Can anyone explain how to go about solving for T?
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Mar15-12, 03:04 AM
P: 761
Log(T) =ln(T)/ln(10) where Log base 10 and ln base e.
T - ln(T)/ln(10) = 1-R
T = -W(X) / ln(10) with X = -ln(10)*(10^(R-1))
W(X) is the Lambert W function.
If the Lambert W function is not implemented on your maths software, you have to use numerical computation in order to solve the equation (Newton-Raphson, or other methods).
The equation has no real solution if R > 1-(1+ln(ln(10))/ln(10) = 0.20349
if R=0.20349 there is only one solution T=1/ln(10) = 0.43429
if 0<R<0.20349 there are two real solutions.

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