How to generate a population of random numbers within a certain exponentially increasing range

This is really more of a question for math.stackexchange, but out of curiosity I provide this solution. Maybe it is sufficient for your needs.

First, ?rexp tells us that it has only two arguments, so we generate a random exponential distribution with the desired length.

set.seed(42)         # for sake of reproducibility
n <- 16068
mr <- c(54850, 150)  # your 'myrange' with less typing

y0 <- rexp(n, 1/25)  # simulate exp. dist.
y <- y0[order(-y0)]  # sort

Now we need a mathematical approach to rescale the distribution.

# f(x) = (b-a)(x - min(x))/(max(x)-min(x)) + a
y.scaled <- (mr[1] - mr[2]) * (y - min(y)) / (max(y) - min(y)) + mr[2]

Proof:

> range(y.scaled)
[1]   150.312 54850.312

That's not too bad.

Plot:

plot(y.scaled, type="l")

Note: There might be some mathematical issues, see therefore e.g. this answer.