Any number can be represented as some other number (called the base) raised to a power. Using 10 as the base, since we already are familiar with scientific notation, we could write:

The logarithm of a number is simply the power (exponent) to which the base must be raised to obtain the given number. Base 10 loagrithms usually are symbolized by the abbreviation log. Taking some examples from the table above, we can write:

log(100) = 2

log(0.001) = -3

log(76) = 1.88

Using the base 10 is a common choice; hence base 10 logarithms often are called common logs.

Another widely used base is the mathematical constant e.

e = 2.7183

Logarithms using the base e are called natural logarithms, and are distinguished from base 10 logarithms by the use of the abbreviation ln.

100 = e^4.605

ln(100) = 4.605

Given the log or ln of a number, we can obtain the number itself by raising the appropriate base to the given power. We call this taking the antilog.

antilog(2) = 10² = 100

antiln(4) = e⁴ = 54.6

Your calculator has buttons labeled:

| ln | - takes the base e logarithm of the displayed number

| log | - takes the base 10 logarithm of the displayed number

The reverse operation, obtaining a number from its logarithm, on most calculators is accomplished by the buttons labeled:

| e^x | - returns the antiln of the displayed number

| 10^x | - returns the antilog of the displayed number

Since the logarithm of a product is the sum of the logs:

log [(x)(y)] = log(x) + log(y)

numbers can be multiplied or divided by adding or subtracting their logarithms. For example, to multiply 13.0 x 4.00:

Or to divide 56.0 by 4.00:

Notice that we are entitled to as many significant figures after the decimal in the logarithm as there were significant digits in the number we took the log of.

Long ago when the dinosaurs ruled the world and I was a boy, precision multliplication and division could be done only by looking up the logs in 10-decimal-place tables that had been compiled laboriously by someone else. In his biography, Neville Shute (author of "On the Beach") tells of commanding a group of "computers" during World War I. These computers were people, whose job was to do the mathematics involved in designing dirigibles, by using 10-place logs. They were subject to a very high frequency of nervous breakdowns. Happily, we don't have to do arithmetic that way anymore, but many relationships in chemistry, physics, biochemistry, geology, and other science and engineering disciplines are logarithmic. Knowing how to manipulate logs therefore is still a very useful skill. For example, in addition to the pH scale we are learning now, the Richter scale for earthquakes is a log scale.