Easy way to compute logarithms without a calculator? I  would need to be able to compute logarithms without using a calculator, just on paper. The result should be a fraction so it is the most accurate. For example I have seen this in math class calculated by one of my class mates without the help of a calculator.   
$$\log_8128 = \frac 73$$
How do you do this?

This answer is additional to awesome answers already given, especially, that of N. F. Taussig.
The definition of a logarithm in reals may help: $\log_b a$ is such a real number $c$ that satisfies $b^c = a$. For example, $\log_2 131072 = 17$ because $2^{17} = 131072$.
Also, you may want to be able to calculate natural logarithms without a calculator. I will tell you a method that I use: since $e^3 \approx 20$, you can take $\ln 20 \approx 3$. Hence, to calculate $\ln n$ in practical applications, first calculate $\log_{20} n$, then multiply it by $3$. Since $20$ is an integer, it's easier to work with. For example, if we need to calculate $\ln 34 627 486 221$, we can do the following:
$$20^8 = 2^8 10^8 = 25 600 000 000\\ \log_{20} 25 600 000 000 \approx 8\\ \ln 25 600 000 000 \approx 8 \cdot 3 = 24\\ \ln 34 627 486 221 = \ln 25 600 000 000 + \ln (34 627 486 221 / 25 600 000 000) \approx 24 + \ln 1.35 \approx 24.35$$
The answer is only 0.13% off, which is very accurate.
Hope this helps!
