TITLE: Why does the logarithm require a special notation?
QUESTION [15 upvotes]: Since the logarithm is the reversed exponentiation, why does it need a distinct notation for it? Why can't we just ask: 
$$2^x=8$$
Instead of:
$$\log_2 8=x$$

REPLY [2 votes]: While pondering the possibility of a less specialized notation, I thought of a specific advantage possessed by the $\log$ notation: it leaves the base of the logarithm implicit.  You can say, for example, that $$\log xy = \log x + \log y$$ rather than this version that is more detailed but no more expressive: $$\text{for all positive $a$,}\  \log_a xy = \log_a x + \log_a y$$
Similarly, we can say that the average running time of the quicksort algorithm is $O(n\log n)$, instead of using the unnecessarily specific $O(n\,\log_2 n)$.
And again, but slightly different: the entropy of a message source is $\sum -p_i\log p_i$.  Here the logarithm is normally taken to be base 2, but it isn't really important, and taking the logarithms to a different base is just changing the units of measurement, which we normally disregard.
Sometimes the base of the logarithm is important, but often, perhaps most often, it isn't, and it's handy to have a notation that sweeps this unimportant detail under the carpet.