Logarithm of Zero

The value of log zero is not defined either in the real or in the complex number system. The logarithm of zero to any base is considered negative infinity. Negative infinity is not defined, so the logarithm of zero is not defined as well.

Consider the logarithm if zero to the base 10:
log 10 0 = x
Converting this to exponential form, you get
10x = 0
Can you think of an exponent, to which when 10 is raised, the result is 0? You need to raise any positive number to a negative exponent if the result is to be a number smaller than it. Since zero is smaller than 10, therefore a negative exponent is required on 10. Thus x belongs to a set of negative numbers. Now, the smaller the result is to be, the more negative should be the exponent. Since zero is the smallest non negative number, therefore the exponent required to be on 10 increases infinitely in the negative direction of the number line. Thus the exponent required to satisfy the above equation is negative infinity.

The above is also true for natural log of zero, and logarithm of zero with any other base.

Thus the value of log 0 is negative infinity, which is not defined.

No comments:

Post a Comment