Log-shift transformation does not squish the data

When I plot a density using geom_spatraster, there's a particularly prominent peak that's 'bright' in the visualization. This spike skews the color scale, compressing the representation of other data points to such an extent that the underlying structure becomes obscured. A standard log transformation doesn't resolve this. My expectation is that a log-shift transformation should compress the data even further. Therefore, I suspect there's something I'm misunderstanding. The following is a minimum working example, using the x-axis to illustrate the issue instead of the 'fill' aesthetic:

library(ggplot2)
library(scales)

# Using mtcars dataset
mtcars[1,"disp"] <- 1500

my_transform <- function(x) log(x + 10)
my_inverse <- function(x) exp(x) - 10

my_trans <-  scales::trans_new("yes_mate", transform = my_transform, inverse = my_inverse)

ggplot(mtcars, aes(x = disp, y = 1,
                   color = disp
                   )) +
  geom_point(size = 5) +
  scale_x_continuous(
    trans = my_trans,
    limits = c(100,1500),
    breaks = c(100, 500, 1000, 1500)) 

So here I would expect the data points to shift further together, but the contrary is the case:

my_transform <- function(x) log(x + 10000)
my_inverse <- function(x) exp(x) - 10000

my_trans <-  scales::trans_new("yes_mate", transform = my_transform, inverse = my_inverse)

ggplot(mtcars, aes(x = disp, y = 1,
                   color = disp
                   )) +
  geom_point(size = 5) +
  scale_x_continuous(
    trans = my_trans,
    limits = c(100,1500),
    breaks = c(100, 500, 1000, 1500)) 

Adding bigger and bigger offsets will bring the values closer together in the absolute scale, but if you similarly scale your axis then counterintuitively the lower values will be bunched together more the larger you make the offset.

x <- c(100, 150, 200, 250, 1500)

par(mfrow=c(1,3), mar=rep(2, 4))
lapply(1:3, \(e) plot(log(x+10^e), x, ann=FALSE))

The solution isn't so much the offset -- the smaller the better but that'll only get you so far. What you want is a log-log(-log? -log?) transform:

## Stack those logs! (more works too)
my_transform <- function(x) log(log(x + 1E-3))
my_inverse <- function(x) exp(exp(x)) - 1E-3

my_trans <-  scales::trans_new("yes_mate", transform = my_transform, inverse = my_inverse)

ggplot(mtcars, aes(x = disp, y = 1, color = disp)) +
  geom_point(size = 5) +
  scale_x_continuous(
    trans = my_trans,
    limits=c(100, 1500),
    breaks = c(100, 500, 1000, 1500)
  )