Based on the mathematics behind what a logarithm is, we find some interesting applications of logarithms. Since a logarithm describes the number of times a common base is a multiple in a product, it only makes sense that the sum of two logarithms is the logarithm of the product of those numbers. For example log(10³)+log(10⁴) would simplify to 3+4 or 7, and notice that the log((10³)*(10⁴)) is log(10⁷) which is also 7. This property also works for division and the subtraction of logarithms. These properties work because you are counting (arithmetic sequence) the number of times a base is found as a multiple in a product (geometric sequence).

There are other properties of logarithms such as ones that involve an argument with an exponent. Using the same values as above, log(10³) is 3, which is also equal to 3*log(10). Again this property is based on the idea of a geometric progression being compared to an arithmetic progression.

An interesting application that was new to me was involving geometric series. Logarithms can be used to find the last term, and also the number of terms of a series with a geometric progression.

To fin the last term in a geometric series, “add the logarithm of the first term to the logarithm of the ratio multiplied by the number of terms less one” and this “sum will be the logarithm of the last term” (Clark, 1843).

To find the number of terms in a geometric series we use a formula for the sum of the terms: S=(arⁿ-a) ⁄(r-1). Thus, the number of terms n=(log(rS-S+a)-log(a))⁄(log(r)). (Clark, 1843)

The system we use to classify our stars today is based on their brightness. Hipparchus, a Greek astronomer was the first to classify the visible stars into six classes. The system Hipparchus used was based on the apparent brightness of the stars. The brightness of a star is referred to as its magnitude. The classification of stars improved in the mid-1800s as the intensity of light flux became measurable through new inventions. (Shirali, 2002).

Fechner's law says: “The response of the senses varies as the logarithm of the stimulus” or “If the difference in apparent magnitudes between two stars is n, then the ratio of their light fluxes is 2.512ⁿ” (Shirali, 2002). If we are considering two stars of magnitude m_A and m_B and light fluxes l_A and l_B then m_A - m_B = 2.512 log₁₀(l_A⁄l_B) (Shirali, 2002).

One of the ways we categorize the intensity of an earthquake is by the Richter scale. The Richter scale is logarithmic: “if r is the Richter magnitude of an earthquake in which the energy released is E, then r = log₁₀E”(Sirali, 2002).

pH means potential for hydrogen. The pH scale is logarithmic in nature and is defined by the equation: pH = -log₁₀[H⁺] where H⁺ refers to the concentration of hydrogen ions. (Shirali, 2002)

There are many other applications for logarithms that I will not go into detail on here. Some of them include: measuring atmospheric pressure, compound interest, measuring sound, radioactivity, populations growth, an the logarithmic spiral.