The main property of a logarithm function is that the logarithm of a product should be the sum of the logarithms of the individual factors:
In his book 'In pursuit of the Unknown', Ian Stewart summarized in a few lines why this equations is so important:
What does it tell us?
How to multiply numbers by adding related numbers instead.
Why is that important?
Addition is much simpler than multiplication.
What did it lead to?
Efficient methods for calculating astronomical phenomena such as eclipses and planetary orbits. Quick ways to perform scientific calculations. The engineers' faithful companion, the slide rule. Radioactive decay and the psychophysics of human perception.
(Ian Stewart, p. 22)
The need for a way to facilitate long operations such as products, divisions and roots is the origin of the invention of logarithms by Napier.
C.H.Edwards Jr. wrote: "The late sixteenth century was an age of numerical computation, as developments in astronomy and navegation called for increasingly accurate and lengthy trigonometric computations. (...) The urgent need, for some device to shorten the labor of tedious multiplications and divisions with many decimal places, was met through the invention of logarithms by Napier and others around the turn of the seventheenth century. " (pag. 142).
Napier's logarithmic tables first appeared in 1614. Henry Briggs published in 1624 the first table of decimal (base 10) logarithms.
We already know that the natural logarithm can be defined as the integral of the equilateral hyperbola:
Using a property of the hyperbola we can probe that:
Now it is easy to probe the main property of logarithm functions:
[Another approach to prove this property is to use the chain rule. For example, see Serge Lang, who wrote: "Please, appreciate the elegance and efficiency of the arguments!", p. 177]
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.
Ian Stewart, In pursuit of the Unknown, 17 Equations that changed the World, Basic Books, 2012.