Getting that X with the Glog function and Lambert’s W

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Facing a simple, yet frustrating formula like this

xe^{ax}=b

and the task to solve it for x left me googling around for hours until I found salvation in Wolfram AlphaWikipedia, and a nice blog post with R-syntax to solve a similar equation.

Using the results from Wolfram Alpha I was able to find the solution with the ‘gsl’ library

# install.packages("gsl")
library(gsl)

# create some example data
dat <- data.frame(a = 0.109861, x = 10)
# a is set so that b is roughly 30. 
# Lazy as I am I used Excel and its solver ability to find numbers

# to check if b is close to 30. Using the initial formula
dat$b <- dat$x * exp(dat$a * dat$x)
dat

# solve for x2 and see if x and x2 are similar and close to 10
dat$x2 <- lambert_W0(dat$a * dat$b)/dat$a
dat

#  a        x  b2       x2
#1 0.109861 10 29.99993 10.00001

# Hurray!

Sometimes life can be so easy (after a long time searching for the right results….).

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