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Value of log 0 is negative infinity (undefined). Because no power can give an answer of zero. it is in fact undefined but written as negative infinity for symbolizing. Otherwise undefined and infinity are two different things.
Log zero is not defined, and if it were defined, it would be more likely to be minus infinity than infinity.
As x tends towards 0 (from >0), log(x) tend to - infinity. As x tends to + infinity so does log (x), though at a much slower rate.
The value of log o is penis
The answer to this whimsical but useless exercise is: Infinity