Definition of the logarithm function as an integral
We define a function log x to be the area under the curve 1/x between 1 and x if x >= 1,
and the negative of the area under the curve 1/x between 1 and x if 0 < x < 1.
Thus log x < 0 if 0 < x < 1 (the graph lies below the x-axis) and log x > 0 if x > 1 (the graph lies above the x-axis).
You can use integral notation:
This function is called the Natural Logarithm or Napierian Logarithm (in honor of their inventor, John Napier 1550-1617, although Napier follow a different approach). In some books and calculators, it is a common practice to use the symbol ln(x).
Remember that of all the integrals of power functions the integral with exponent -1 was the only one which we cannot evaluate.
The Fundamental Theorem of Calculus in this case says that
(You can see a basic and nice demonstration of this property in Serge Lang, p. 176).
If you change x (you are modifying the upper limit of integration) you can see the tangent to the logarithm function in this point and how the derivative in this point is its inverse (then the value is on the equilateral hyperbola). [The derivative is about as simple as one could ask! Spivak]
A very important property we want logarithms to have is that the logarithm of a product should be the sum of the logarithms of the individual factors
We can deduce this property using the chain rule (see Serge Lang, p. 177, or Spivak, p. 315, for example). And Serge Lang wrote: "Please appreciate the elegance and efficiency of the arguments!".
But we are going to see that a property of the integral of the equilateral hyperbola justify the property of the logarithm function.
[This relation between the logarithm and the rectangular hyperbola xy=1 was discovered by the Belgian Jesuit Gregory St. Vincent in 1647]
The function log is clearly increasing (we defined it as an area). It has a positive derivative everywhere so it is strictly increasing.
For small values of x, the slope is large, and moreover, it increases without bound as x decreases towards zero.
The curve has slope 1 when x=1. For x > 1, the slope gradually decreases toward zero as x increases indefinitely.
Since the derivative becomes very small as x becomes large, log consequently grows more and more slowly.
It is not clear if log is bounded or not. It might be suspected that the values of log have and upper bound but you can prove that log is unbounded above nor below. Since log is continuous, it actually takes on all values. More than that, for every real number a there is one positive real number r whose logarithm is a.
Then we can define an inverse function. The domain of this inverse function is all real numbers. This important function is the Exponential function.
There is exactly one number whose natural logarithm is equal to 1. This is a very importan number, like pi, and we use a special symbol due to Leonard Euler (1707-1783), e. He was the first to recognize the importance of this number.
A. I. Markushevich, Areas and Logarithms, D.C. Heath and Company, 1963.
Serge Lang, A First Course in Calculus, Third Edition, Addison-Wesley Publishing Company.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.
Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Otto Toeplitz, The Calculus, a genetic approach, The University of Chicago Press, 1963.
Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, Springer-Verlag New York Inc., 1980.
C.H. Edwards, Jr., The Historical Development of the Calculus, Springer-Verlag New York Inc., 1979.