Log-transforming a value less than 1 results in a negative number. Here is a quick demonstration in R:
# Generating a sequence of values between 0 and 1 (inclusive), incrementing by 0.1
x <- seq(0, 1, by = 0.1)
x
[1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
# Taking the log() of each value in this sequence
round(log(x), 2)
[1] -Inf -2.30 -1.61 -1.20 -0.92 -0.69 -0.51 -0.36 -0.22 -0.11 0.00
It is entirely possible for the sign of a Pearson's correlation to flip as a consequence of logging observations between 0 and 1. In such contexts, it's possible for $\rho(X,Y)$ to flip sign if we $\log X$ or $\log Y$, or both. See below for a quick-and-dirty simulation. In each iteration, I assessed the correlation between 10 pairs of values. The correlation was preserved in 7 out of the 10 trials. Note, upon each draw of 10 observations, a subset fall between 0 and 1, which reverse in sign after the log transformation; this has the potential to influence the relationship between variables.
set.seed(13)
sims <- 10
levels <- vector(mode = "numeric", length = 10) # storage for the numeric values in levels
logged <- vector(mode = "numeric", length = 10) # storage for the "logged" numeric values
for (i in 1:sims) {
x <- exp(rnorm(10)); y <- exp(rnorm(10)) # drawing 'positive' random deviates
levels[i] <- cor(x, y) # correlation in levels
logged[i] <- cor(log(x), log(y)) # correlation of the logged values
}
flipped <- (levels > 0 & logged < 0) | (levels < 0 & logged > 0)
preserved <- !flipped
# Were the correlations between pairs preserved?
cbind(levels, logged, flipped, preserved)
levels logged flipped preserved
[1,] 0.10728307 -0.01369591 1 0
[2,] 0.93652958 0.62956976 0 1
[3,] 0.01300703 0.07601658 0 1
[4,] -0.06794333 0.37656387 1 0
[5,] 0.17978986 0.27654877 0 1
[6,] 0.19476326 0.54571601 0 1
[7,] -0.09462134 -0.04706490 0 1
[8,] -0.25225396 -0.40215868 0 1
[9,] -0.20694668 -0.05695933 0 1
[10,] -0.04839709 0.08447948 1 0
Another example is when a transformation dampens an outlier that was exerting major leverage. Review the top answer here for a very simple and clear demonstration of this. If you're working with per capita rates (e.g., limited number of incidents per unit population) or proportions, then a log transformation can influence or even reverse the sign of a coefficient.