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?

To evaluate $\log_8 128$, let 
$$\log_8 128 = x$$
Then by definition of the logarithm, 
$$8^x = 128$$
Since $8 = 2^3$ and $128 = 2^7$, we obtain
\begin{align*}
(2^3)^x & = 2^7\\
2^{3x} & = 2^7
\end{align*}
If two exponentials with the same base are equal, then their exponents must be equal.  Hence,
\begin{align*}
3x & = 7\\
x & = \frac{7}{3}
\end{align*}
Check:  If $x = \frac{7}{3}$, then 
$$8^x = 8^{\frac{7}{3}} = (8^{\frac{1}{3}})^7 = 2^7 = 128$$
