Using negative number
For any value less than or equal to 0 , we will get ValueError

import math
print(math.log1p(-2))

Above code will generate error.

math.log1p(x) also works for negative values of x as long as 1 + x > 0. This is useful for handling small negative changes in probability models.

import math
x = -0.001 # 0.1% decrease
log_value = math.log1p(x)
print(log_value) # -0.0010005003335835335

Precision with log1p() vs log(1 + x)

When x is very small (close to 0), using math.log1p(x) is preferred over math.log(1 + x) due to the precision issues that arise when adding small numbers to 1. Here's how the two functions differ:

import math
x = 1e-10
print(math.log1p(x)) # High precision for small x
print(math.log(1 + x)) # Slight loss of precision

Explanation:
For very small values of x, log1p(x) provides more accurate results because it directly computes the natural logarithm of 1 + x without actually adding x to 1, which avoids losing precision due to floating-point arithmetic.

Real-World Use Case: Finance and Probability

In finance, log1p() can be used to compute the natural log of growth rates, where small percentage changes (like 0.01%) are frequent. Similarly, in probability and statistics, log1p() helps calculate log-probabilities when values are very close to 0.

# Log of a small percentage increase in a financial model
growth_rate = 0.001 # 0.1% growth
log_growth = math.log1p(growth_rate)
print(log_growth)

This approach is more accurate than using math.log(1 + growth_rate) and ensures that small growth rates are calculated correctly without precision loss.

Use Case in Machine Learning:

In machine learning, log1p() can be used when calculating the logarithm of small probability values in models like logistic regression. Using log1p() avoids underflow or precision issues when probabilities are very close to 0 or 1.

import math
# Logarithm of a small probability value in a logistic regression model
probability = 0.0001
log_prob = math.log1p(-probability)
print(log_prob)

This ensures accurate computation for small probability values when modeling predictions.