Logarithms
The logarithm of a positive real number x to the base-a is the number y that satisfies the equation ay = x. In symbols, the logarithm of x to the base-a is loga x, and, if ay = x, then y = loga x.
Essentially, the logarithm to base-a is a function: To each positive real number x, the logarithm to base-a assigns x a number y such that ay = x. For example, 102 = 100; therefore, log10 100 = 2. The logarithm of 100 to base-10 is 2, which is an elaborate name for the power of 10 that equals 100.
Any positive real number except 1 can be used as the base. However, the two most useful integer bases are 10 and 2. Base-2, also known as the binary system, is used in computer science because nearly all computers and calculators use base-2 for their internal calculations. Logarithms to the base-10 are called common logarithms. If the base is not specified, then base-10 is assumed, in which case the notation is simplified to log x.
Some examples of logarithms follow.
log 1 = 0 because 100 = 1
log 10 = 1 because 101 = 10
log 100 = 2 because 102 = 100
log2 8 = 3 because 23 = 8
log2 2 = 1 because 21 = 2
log5 25 = 2 because 52 = 25
log3 = −2 because 3-2 =
The logarithm of multiples of 10 follows a simple pattern: logarithm of 1,000, 10,000, and so on to base-10 are 3, 4 and so on. Also, the logarithm of a number a to base-a is always 1; that is, loga a = 1 because a1 = a.
Logarithms have some interesting and useful properties. Let x, y, and a be positive real numbers, with a not equal to 1. The following are five useful properties of logarithms.
1. loga(xy) = loga x + logay, so log10(15) = log10 3 + log10 5
2. loga = loga x − loga y, so log (⅔) = log 2 − log 3
3. loga xr = r loga x, where r is any real number, so log 35 = 5 log 3
4. loga = −loga x, so log (¼) = (−1) log 4 because ¼ = (4)-1
5. loga ar = r, so log10 103 = 3
Logarithms are useful in simplifying tedious calculations because of these properties.
History of Logarithms
The beginning of logarithms is usually attributed to John Napier (1550–1617), a Scottish amateur mathematician. Napier's interest in astronomy required him to do tedious calculations. With the use of logarithms, he developed ideas that shortened the time to do long and complex calculations. However, his approach to logarithms was different from the form used today.
Fortunately, a London professor, Henry Briggs (1561–1630) became interested in the logarithm tables prepared by Napier. Briggs traveled to Scotland to visit Napier and discuss his approach. They worked together to make improvements such as introducing base-10 logarithms. Later, Briggs developed a table of logarithms that remained in common use until the advent of calculators and computers. Common logarithms are occasionally also called Briggsian logarithms.
