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Why do we use log?

"Logarithm" is a word made up by Scottish mathematician John Napier (1550-1617), from the Greek word logos meaning "proportion, ratio or word" and arithmos meaning "number", ... which together makes "ratio-number" !

In its simplest form logarithm gives the info about:
           how many a number can give another number


basically its used to get the power ie. the exponent of the base number in the log with respect to the log number

for eg:

 1. log2(8)=3   ie  23 = 8

 how many a number (to the base number) (2)
another number (log number) (8)
raise to power (3)

2.  log2(4)=2   ie  22 = 4 


same goes for the negative logarithms:

1.   \log_2 \!\left( \frac{1}{2} \right) = -1,\,
                since
          2^{-1} = \frac 1 {2^1} = \frac 1 2.


reference: 
www.mathsisfun.com/algebra/logarithms.html



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