alternative definition of the natural logarithm
The natural logarithm function (http://planetmath.org/NaturalLogarithm2) can be defined by an integral, as shown in the entry to which this entry is attached. However, it can also be defined as the inverse function of the exponential function .
In this entry, we show that this definition of yields a function that satisfies the logarithm laws and hold for any positive real numbers and and any real number . We also show that is differentiable with respect to on the interval with derivative . Note that the logarithm laws imply that . The mean-value theorem implies that these properties characterize the logarithm function.
The proof of the first logarithm law is straightforward. Let and be positive real numbers. Then using the fact that and are inverse functions, we find that
Since is an injective function, the equation implies the first logarithm law.
For the second logarithm law, observe that
|Title||alternative definition of the natural logarithm|
|Date of creation||2013-03-22 16:11:10|
|Last modified on||2013-03-22 16:11:10|
|Last modified by||CWoo (3771)|