Help to figure out equation solution

  1. Hey,

    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?
  2. jcsd
  3. 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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