The C programming log10 function is defined in the math.h header file and is used to calculate the logarithm of a given value x to the base 10. The logarithm of a number to the base 10 is the power to which 10 must be raised to obtain the given number.

The log10 function takes a single argument of type double, which represents the value to be logged. The function returns a double value, which represents the calculated logarithm to the base 10.

Here's an example usage of the log10 function to calculate the logarithm to the base 10 of a given input:

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

int main() {
    double x = 1000;
    double log10_value = log10(x);
    
    printf("The logarithm to the base 10 of %f is %f\n", x, log10_value);
    
    return 0;
}

In this example, the log10 function is used to calculate the logarithm to the base 10 of 1000, which is stored in the x variable. The calculated logarithm to the base 10 is then stored in the log10_value variable, and printed to the console using the printf function.

Note that the log10 function can also be used to calculate the logarithm of a value to a different base by using the change of base formula, which states that log_b(x) = log_a(x) / log_a(b), where a and b are the bases of the logarithms.