log1p()

log1p(x)

Returns: numeric · Updated March 13, 2026 · Base Functions

math log log1p numerical-stability base

The log1p() function computes log(1 + x), the natural logarithm of one plus the input. It is more accurate than log(1 + x) when x is very small.

Syntax

log1p(x)

Parameters

Parameter	Type	Default	Description
`x`	numeric	—	A numeric vector or scalar

Examples

Basic usage

# log1p(0) = log(1) = 0
log1p(0)
# [1] 0

# log1p(1) = log(2)
log1p(1)
# [1] 0.6931472

# For very small values, log1p is more accurate
x <- 1e-10
log(1 + x)
# [1] 0

log1p(x)
# [1] 1e-10

Numerical stability demonstration

# When x is very small, log(1 + x) loses precision
x <- 1e-15
log(1 + x)
# [1] 0  # precision lost!

log1p(x)
# [1] 1e-15  # accurate

# Relative error comparison
x <- 1e-8
(log(1 + x) - x) / x  # relative error in log()
# [1] -5.000001e-08

(log1p(x) - x) / x   # relative error in log1p()
# [1] 0

Common Patterns

Statistical calculations

# Log-likelihood with small probabilities
p <- 1e-10
log(p + 1)  # loses precision
log1p(p)    # accurate

# Geometric mean for values near 1
x <- c(1.001, 1.002, 1.003)
exp(mean(log1p(x - 1)))
# [1] 1.002

Financial calculations

# Continuous compounding for small rates
rate <- 0.0001
time <- 1
log1p(rate * time)
# [1] 0.000099995