Hi @zoep,
One simple approach is adding a constant to the data. For example, you could add 1 to every point, then log transform.
x <- c(0, 1, 2, 3)
log(x)
log(x + 1)
As a follow-up there is a package called bestNormalize which will test out multiple "normalizing" transformations for you, you can try that if you want. If you inspect the log_x transformation it does exactly as I described, adds a small constant to every data point (max(0, -min(x) + eps)).
library(bestNormalize)
x <- rexp(100)
hist(x)
xtrans <- bestNormalize(x)
hist(xtrans$x.t)