By the early 1600s, astronomy had progressed to the point of finding accurate information about the motion of the heavenly bodies.

Also, navigation had led to a more systematic exploration of Earth. In making the accurate measurements needed in astronomy and navigation, many lengthy calculations had to be performed, and all such calculations had to be done by hand.

Noting that astronomers’ calculations usually involved sines of angles, John Napier (1550–1617), a Scottish mathematician, constructed a table of values that allowed multiplication of these sines by addition of values from the table. These tables of logarithms first appeared in 1614. Therefore, logarithms were essentially invented to make multiplications by means of addition, thereby making them easier.

Napier’s logarithms were not in base 10, and the English mathematician Henry Briggs (1561–1631) realized that logarithms in base 10 would make the calculations even easier. He spent many years laboriously developing a table of base 10 logarithms, which was not completed until after his death.

Logarithms were enthusiastically received by mathematicians and scientists as a long-needed tool for lengthy calculations. The great French mathematician Pierre Laplace (1749–1827) stated “by shortening the labors doubled the life of the astronomer.” Logarithms were commonly used for calculations until the 1970s when the scientific calculator came into use.

In this chapter, we study the logarithmic function and the exponential function. Although logarithms are no longer used directly for calculations, they are of great importance in many scientific and technical applications and in advanced mathematics. For example, they are used to measure the intensity of sound, the intensity of earthquakes, the power gains and losses in electrical transmission lines, and to distinguish between a base and an acid. Exponential functions are used in electronics, mechanical systems, thermodynamics, nuclear physics, biology in studying population growth, and in business to calculate compound interest.