Cool factorials!

My python code(names of variables are chosen randomly. ) -:

import math

n = 1000

m = 100

k = 10

s = math.factorial(n)

j = math.factorial(m)

k = math.factorial(k)

In = len(str(s))

In1 = len(str(j))

In2 = len(str(k))

print(In + ln1 + ln2)

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import math
def digits(n):
    return int(math.log10(math.factorial(n))) + 1

print digits(1000) + digits(100) + digits(10)
2733

Seems like everyone is using programming to solve this problem. Is there a more mathematical way to solve this problem without relying on computers?

MATLAB Code:

f=@(x)floor(1+sum(log10(x)));
answer=f(1:1000)+f(1:100)+f(1:10);

Here is another way to solve it (in Java) -

public static void main(String[] args) throws IOException{
    System.out.println(fact((long) 10) + fact((long) 100) + fact((long) 1000));
}

private static long fact(long n) {
    double logFacN = 0;
    for (long i = 2; i <= n; i++)
        logFacN += Math.log10(i);
    return (long) logFacN + 1;
}

Python:

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from math import factorial
def xi(n):
    return len(str(factorial(n)))
print xi(1000) + xi(100) + xi(10)

$\xi(1000) = 2568$

$\xi(100) = 158$

$\xi(10) = 7$

$2568+158+7=2733$