I am not looking for help with my homework. I just need someone to show me the direction to do it. I know the answer theoretically. I just stuck with idea of how to prove it mathematically. here is the question.
Representing a number in the octal system require, on the average, about 10 percent more characters than in the decimal system.
How can I prove this mathematically?
Suppose you wanted to represent a given number x in both systems. In the decimal system, this will take in the order of log10(x) digits. In the octal system, it will take in the order of log8(x) digits.
For any a and b, loga(b) can be written as logc(b)/logc(a) for a given c. In particular, let c=10. Therefore, log8(x) = log10(x)/log10(8) ~= 1.1 log10(x), which means log8(x) is about 1.1 times greater than log10(x) for any given x. Note that this result is exact aside from the rounding. What is not exact is approximating the number of digits by log10(x) and log8(x).