Help to figure out equation solution

sj21

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?

JJacquelin

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.