Derivative of a Logarithm




We start with the function for the logarithm and calculate a change in “y” caused by a change in “x” as





where we have defined a new variable



The derivative operation for the logarithm function is



where we have defined the new variable



Recalling the binominal theorem, we can write





where the “binominal coefficients” are the combinatorials



noting the last expression above has “j” terms in the numerator.  But in any event using this expression we can write



And the limit can be expressed as



The derivative of the logarithm is thus



or for the special case of the natural logarithm






A logarithm is simply an exponent.   If we begin with the following expression



then we can define the logarithm as follows



In simple parlance, we say that the logarithm is that exponent which when applied to a base, yields a particular value.


A trivial result from the definition of a logarithm gives us the following



or simply



The consequences of this simple definition are several, as follows