# The Logarithm and its Identities

**Episode #2 of the course “Equations that changed the world”**

In the days before computers, mathematical computations were long and tedious. Mathematicians experimented with shortcuts to make calculations of incredibly large and incredibly small numbers more efficient and more accurate. It was in the early 17th century that John Napier invented the concept called a “logarithm,” named for the Greek words meaning “proportionate number” or “number in proportion.”

The concept and its new mathematical computations were an immediate success and highly popular, embraced by mathematicians around the world. By the end of the 17th century, the concept of the logarithm had been expanded and connected to geometry and trigonometry, as well as the properties of exponents, which had been used since the 3rd century BCE.

Logarithms summarize and compute numbers related to one another by exponential patterns. Once mathematicians know the interval by which the numbers are related, they can establish a “base” to compute any number in that relationship. For example, scientists measure earthquakes on a scale that defines the power of the earthquake in comparison with other earthquakes, in increments of 10. Therefore, an earthquake measured as “3.0” is 10 times more powerful than an earthquake measured as “2.0.” This scale is a logarithm with a base of 10, describing relationships between earthquakes’ power as exponential.

Three main types of logarithms are used today. The Common Logarithm, or logarithmic formula with a base of 10, is also commonly called the Briggsian logarithm. It is most often used in engineering, biology, and astronomy. The most common logarithm for computer science and engineering is the Binary Logarithm, which operates with a base of 2. It is used in music theory and photography, in addition to information technologies. Lastly, the Natural Logarithm is commonly used in mathematics, physics, economics, statistics, and other fields. The Natural logarithm uses a base of “*e*,” which is an irrational constant equal to approximately 2.718.

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