Q.Evaluate $\log_{2}64$ without using tables.

The logarithm of a number $N$ to a base $a$ is the power to which $a$ must be raised to give $N$: $a^{x}=N \Leftrightarrow \log_{a}N = x$, valid for $a>0,\,a\neq 1,\,N>0$. Exponential and logarithmic forms are two ways of writing the same relationship, and fluency in converting between them underlies everything else. Basic values follow at once from the definition — $\log_{a}1 = 0$, $\log_{a}a = 1$, and $\log_{a}(a^{n}) = n$ — while logarithms of zero and of negative numbers are undefined. Three laws do the real work: the product law $\log(MN)=\log M+\log N$, the quotient law $\log(M/N)=\log M-\log N$, and the power law $\log(M^{p})=p\log M$ (which also handles roots as fractional powers). The change-of-base rule $\log_{a}N = \log_{b}N/\log_{b}a$ lets any logarithm be evaluated through base-10 tables, with the corollary $\log_{a}b\times\log_{b}a = 1$. Together these turn multiplication into addition, division into subtraction and powers into multiplication.