Why is a log transformation commonly applied to skewed numerical features before modeling, mathematically?

Many real-world quantities — income, population, word frequencies, prices — follow a right-skewed (long right tail) distribution, often approximately log-normal. The mathematical property of the logarithm that makes it useful here is that it compresses large values much more than small ones: log(1000) - log(100) ≈ 2.3 while log(100) - log(10) ≈ 2.3 as well — equal ratios become equal differences after a log transform. This converts multiplicative relationships into additive ones and pulls in the long tail, making the distribution closer to symmetric/normal.

This matters for linear models because OLS assumes residuals are normally distributed with constant variance (homoscedasticity); a skewed target or feature violates this and can lead to heteroscedastic residuals where prediction error grows with the magnitude of the target. It also matters for distance-based and gradient-based methods, where a few extreme outliers in the raw scale would otherwise dominate the loss or distance calculations.

import numpy as np
from sklearn.preprocessing import FunctionTransformer, PowerTransformer
import pandas as pd

# Simulating right-skewed income data
income = np.random.lognormal(mean=10, sigma=1, size=1000)
print('Skewness before:', pd.Series(income).skew())     # highly positive

log_income = np.log1p(income)  # log1p handles zero values safely: log(1+x)
print('Skewness after:', pd.Series(log_income).skew())  # close to 0

# Integrate into a scikit-learn pipeline
log_transformer = FunctionTransformer(np.log1p, validate=True)
X_log = log_transformer.fit_transform(X[['income']])

# Box-Cox / Yeo-Johnson: more general power transforms that
# find the optimal transformation parameter automatically
pt = PowerTransformer(method='yeo-johnson')  # handles negative values too
X_transformed = pt.fit_transform(X)