Q.Simplify and evaluate: $2\log 5 + \log 8 - \log 2$ (all to base 10).

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 …