How does using log10 correctly calculate the length of a integer?

math logarithm
user2926999 picture user2926999 · Jun 12, 2014 · Viewed 8.5k times · Source 
int length = (int) floor( log10 (float) number ) + 1;

My question is essentially a math question: WHY does taking the log10() of a number, flooring that number, adding 1, and then casting it into an int correctly calculate the length of number?

I really want to know the deep mathematical explanation please!

Answer

Yu Hao picture Yu Hao · Jun 12, 2014

For an integer number that has n digits, it's value is between 10^(n - 1)(included) and 10^n, and so log10(number) is between n - 1(included) and n. Then the function floor cuts down the fractional part, leaves the result as n - 1. Finally, adding 1 to it gives the number of digits.

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