Definition

The log function computes the natural logarithm (base $e \approx 2.71828$) of a floating-point number. It is part of the C standard math library and implements IEEE 754 compliant logarithmic evaluation. Unlike log10 or log2, log exclusively calculates logarithms in base $e$, making it essential for continuous growth/decay models, entropy calculations, and mathematical transformations.

Syntax and Parameters

#include <math.h>
double log(double x);
float logf(float x);          // C99
long double logl(long double x); // C99

Parameter	Type	Description
`x`	Floating-point	The input value. Must be strictly greater than `0.0` for a finite real result

Return Value and Special Cases

Returns $\ln(x)$ as a floating-point value. IEEE 754 and C standard define precise behavior for boundary conditions:

log(1.0) → 0.0
log(0.0) → -HUGE_VAL (negative infinity)
log(x < 0) → NaN (domain error)
log(+inf) → +inf
log(NaN) → NaN
Error Reporting:
x < 0 → errno set to EDOM
x == 0 → errno set to ERANGE (pole error)

Code Examples

#include <stdio.h>
#include <math.h>
#include <errno.h>
int main(void) {
// Basic usage
double val1 = log(2.718281828);  // ≈ ln(e) = 1.0
double val2 = log(10.0);         // ≈ 2.302585
double val3 = log(1.0);          // 0.0
printf("ln(e): %.6f\n", val1);
printf("ln(10): %.6f\n", val2);
printf("ln(1): %.6f\n", val3);
// Error handling for invalid domain
errno = 0;
double invalid = log(-5.0);
if (errno == EDOM || isnan(invalid)) {
printf("Error: log of negative number is undefined in real space\n");
}
// Handling zero (pole error)
errno = 0;
double zero_case = log(0.0);
if (errno == ERANGE || isinf(zero_case)) {
printf("log(0) = -infinity\n");
}
return 0;
}

Rules and Constraints

Domain Restriction: Defined only for $x > 0$. Negative inputs are mathematically undefined in real arithmetic and return NaN.
Type Specificity: Implicit integer-to-double conversion occurs if integers are passed. Very large integers may lose precision before evaluation.
Precision Limits: Results are accurate to within a few ULPs. Near $x = 1.0$, standard log suffers from catastrophic cancellation due to floating-point representation.
math_errhandling Macro: Determines whether errors are signaled via errno, floating-point exceptions, or both. Default behavior varies by implementation.

Best Practices

Validate domain before calling: if (x <= 0.0) { handle_error(); }
Use log1p(x) for values near 1.0: log1p(y) computes $\ln(1 + y)$ accurately when $y \approx 0$. Avoid log(1.0 + y) due to precision loss.
Prefer base-specific functions: Use log2(x) or log10(x) instead of log(x) / log(base) for accuracy and performance.
Match precision: Use logf or logl when working with float or long double to avoid silent conversions.
Check error states: Use isnan(), isinf(), and errno before propagating results to downstream calculations.
Document expected ranges: Logarithmic functions amplify numerical instability near boundaries. Clearly specify valid input domains.

Common Pitfalls

🔴 Confusing log with log10: log is natural logarithm, not base 10. Misuse leads to incorrect scaling in scientific calculations.
🔴 Ignoring domain errors: Passing x <= 0 silently produces NaN or -inf, corrupting entire mathematical pipelines.
🔴 Precision loss near 1.0: log(1.0000000000000002) may evaluate to 0.0 due to floating-point cancellation. Use log1p() instead.
🔴 Integer division in base conversion: log(x) / log(2) with integer literals triggers integer division before conversion, yielding 0 or garbage.
🔴 Missing math library: POSIX systems require -lm during compilation. Omission causes undefined reference to 'log' linker errors.
🔴 Assuming exact inverses: exp(log(x)) rarely equals x exactly due to rounding errors. Use tolerance-based comparisons for equality checks.

Performance and Alternatives

Hardware Acceleration: Modern FPUs provide dedicated logarithmic instructions (e.g., fyl2x on x86). log typically maps directly to these.
log1p(x): Highly optimized for small $x$. Uses polynomial approximations that preserve significant digits lost in 1.0 + x.
Base Conversion: log2() and log10() (C99) are faster and more accurate than manual division.
Approximation Tables: For embedded or real-time systems without FPU support, lookup tables with linear interpolation offer deterministic latency at the cost of memory.
Vectorization: SIMD math libraries (Intel SVML, ARM Ne10, OpenLibm) provide batched log evaluation for array processing.

Linking Requirements

On POSIX-compliant systems (Linux, macOS, Unix), mathematical functions reside in a separate library. Compile with:

gcc main.c -lm -o main

Windows MSVC and modern compilers with standard libraries often link math functions automatically, but explicit -lm remains required for cross-platform portability and strict standard compliance.

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