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

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 Alpha, Wikipedia, 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….).

# Appendix: Improvements

After revisiting this article some time later, I wondered what the speed is compared to Dan Kelley’s (see comment below) alternative. After firing up some repetitions using microbenchmark I got the following:

```library(gsl)
library(rootSolve)
library(microbenchmark)
library(ggplot2)

dat <- data.frame(a = 0.109861, x = 10)
dat\$b <- dat\$x * exp(dat\$a * dat\$x)

f <- function(x, a, b) x*exp(a*x) - b

autoplot(microbenchmark(
lambertW = dat\$x2 <- lambert_W0(dat\$a * dat\$b)/dat\$a,
uniroot = dat\$x3 <- uniroot.all(f, interval = c(0, 100), a = dat\$a, b = dat\$b),
times = 10000))
``` ## 3 thoughts on “Getting that X with the Glog function and Lambert’s W”

1. […] Getting that X with the Glog function and Lambert’s W 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 Alpha, Wikipedia, and a nice blog post with R-syntax to solve a similar equation. […]

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2. Dan Kelley says:

Is the following helpful?

> a b f library(rootSolve)
> uniroot.all(f, interval=c(0,100))
 10

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3. David Zimmermann says:

Hi Dan,
thanks for the comment, I haven’t looked into the rootSolve library. Can you post a minimum working example?

Thanks!

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