C Math log10() Function

What You’ll Learn

The log10() function computes the common logarithm log₁₀(x)—the logarithm to base 10. It answers “how many times do I multiply 10 to get x?” and appears in decibels, scientific notation, and orders of magnitude.

Definition and Usage

The common logarithm answers: “To what power must 10 be raised to get x?” If 10t = x, then log10(x) = t.

In C, log10(x) requires x > 0. log10(1) = 0 because 10&sup0; = 1. log10(10) = 1 because 10¹ = 10. log10(1000) = 3 because 10³ = 1000.

📝 Syntax

Standard C declaration:

double log10(double x);

Related variants

float log10f(float x);             /* <math.h> */
long double log10l(long double x); /* <math.h> */

Parameters

• x — positive value; domain is x > 0.

Return Value

• Base-10 logarithm log₁₀(x) as double.
• log10(1000.0) returns 3.0.
• log10(0) returns −∞; log10(negative) returns NaN.

Headers and linking

• #include <math.h>
• Compile: gcc log10.c -std=c11 -o log10 -lm
• Change of base: log10(x) = log(x) / log(10.0)

Examples Gallery

Compile every example with gcc file.c -std=c11 -o out -lm. Base-10 logs match how we count digits and powers of ten in everyday science.

#include <math.h>
#include <stdio.h>

int main(void) {
    double number = 1000.0;
    double result = log10(number);

    printf("Log base 10 of %.0f = %.0f\n", number, result);

    return 0;
}

Log base 10 of 1000 = 3

#include <math.h>
#include <stdio.h>

int main(void) {
    printf("log10(1)  = %.1f  (10^0 = 1)\n", log10(1.0));
    printf("log10(10) = %.1f  (10^1 = 10)\n", log10(10.0));

    return 0;
}

log10(1)  = 0.0  (10^0 = 1)
log10(10) = 1.0  (10^1 = 10)

#include <math.h>
#include <stdio.h>

int main(void) {
    double values[] = { 7.0, 42.0, 999.0, 1000.0, 12345.0 };
    size_t i;

    printf("   n       log10(n)   digits (floor+1)\n");
    for (i = 0; i < 5; i++) {
        double n = values[i];
        int digits = (int)floor(log10(n)) + 1;
        printf("%8.0f  %8.4f   %d\n", n, log10(n), digits);
    }

    return 0;
}

   n       log10(n)   digits (floor+1)
       7    0.8451   1
      42    1.6232   2
     999    2.9996   3
    1000    3.0000   4
   12345    4.0915   5

#include <math.h>
#include <stdio.h>

int main(void) {
    double power = 100.0;
    double reference = 1.0;
    double db = 10.0 * log10(power / reference);

    printf("Power ratio: %.0f : %.0f\n", power, reference);
    printf("Level: %.1f dB\n", db);

    return 0;
}

Power ratio: 100 : 1
Level: 20.0 dB

#include <math.h>
#include <stdio.h>

int main(void) {
    double x = 100.0;

    printf("x = %.0f\n", x);
    printf("log10(x) = %.4f  (base 10)\n", log10(x));
    printf("log(x)   = %.4f  (base e)\n", log(x));
    printf("log(x)/log(10) = %.4f  (change of base)\n", log(x) / log(10.0));

    return 0;
}

x = 100
log10(x) = 2.0000  (base 10)
log(x)   = 4.6052  (base e)
log(x)/log(10) = 2.0000  (change of base)

🚀 Common Use Cases

• Scientific notation — express large/small numbers as mantissa × 10^exponent.
• Decibels (dB) — compress wide power or amplitude ratios.
• Orders of magnitude — compare scales (Richter, stellar brightness).
• Digit estimation — floor(log10(n))+1 for positive integers.
• Inverse of 10^t — solve 10^t = x with t = log10(x).

📝 Notes

• Common log uses base 10, not base e.
• Domain: x > 0 only. Zero and negatives are errors.
• log10(1) = 0; log10(10) = 1; log10(100) = 2.
• Calculator log often means base 10; C log() means natural log.
• Change of base: log10(x) = log(x) / log(10.0).
• Link with -lm on GCC/Clang Unix-like builds.

⚡ Optimization

log10() is implemented in the platform math library. Prefer log10(x) over log(x) / log(10.0) when you need base 10—it is clearer and may be faster. In hot loops, profile the overall algorithm first; use log10f when float precision is enough.

💡 Best Practices