Sunday, December 27, 2015

A zero to hero...................


Chief among them are: the logarithm of the number one is zero; and the logarithm as x approaches zero from the right is negative infinity. What makes natural logarithms unique is to be found at the single point where all logarithms are zero, namely the logarithm of the number one. At that specific point the "slope" of the curve of the graph of the natural logarithm is also precisely one. Logarithms to a higher base than e, such as those to the base 10, exhibit a slope at that point less than one, while logarithms to a lower base than e, such as those to the base 2, exhibit a slope at that point greater than one. While the methods for computing the "value" of e are fascinating from various mathematical perspectives, they all can be thought of as resulting from the pursuit of this condition. Another way of conceptualizing this is to realize that, for any numeric value close to the number one, the natural logarithm can be computed by subtracting the number one from the numeric value. For example, the natural logarithm of 1.01 is 0.01 to an accuracy better than 5 parts per thousand. With similar accuracy one can assert that the natural logarithm of 0.99 is minus 0.01. The accuracy of this concept increases as one approaches the number one ever more closely, and reaches completeness of accuracy precisely there. To the same extent that the number one itself is a number common to all systems of counting, so also the natural logarithm is independent of all systems of counting. In the English language the term adopted to encapsulate this concept is the word "natural".

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