Log zero is not defined, and if it were defined, it would be more likely to be minus infinity than infinity.
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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.
Zero times infinity is defined as "indeterminate".
Because zero multiplied by any number is always zero, but anything multiplied by infinity is infinity. Zero times infinity is being pulled both ways. Also, the definition of infinity is any number x divided by 0. When you multiply zero by infinity, the zeroes "cancel out", leaving absolutely nothing behind.
infinite
No. Zero multiplied or divided by anything is zero.