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1.18 Logarithms

In addition to scientific notation, logarithms are often used to express and manipulate large and small numbers. The common logarithm, or logarithm to the base 10, of a number is written:

If x = 10y, then y = log10x, or y = log x

The ln/e^x scientific calculator key. Click here for original source URL.

The logarithm defines a compressed scale that is very handy in astronomy where numbers with an enormous range must often be compared. For short, they are also called "logs." On a calculator, you can take the logarithm of a number by entering the number and pressing the button marked LOG or LOG10. The reverse operation, raising a number to a power of ten, is done by entering the power of ten and pressing the button marked ALOG or ALOG10. Note that the power of ten does not have to be an integer or a positive number. The list below has the logarithms of integers from 1 to 10, along with the fractional powers of ten from 0.1 to 1. These conversions are quoted with three significant figures although you calculator can give them with much higher precision.

The list also shows the very simple form of the logarithms of powers of ten.

• x = 1, log x = 0.000

• x= 2, log x = 0.301

• x= 3, log x = 0.477

• x= 4, log x = 0.602

• x= 5, log x = 0.699

• x= 6, log x = 0.788

• x= 7, log x = 0.845

• x= 8, log x = 0.903

• x= 9, log x = 0.954

• x= 10, log x = 1.000

• log x = 0.1, x = 1.259

• log x = 0.2, x = 1.585

• log x = 0.3, x = 1.995

• log x = 0.4, x = 2.512

• log x = 0.5, x = 3.162

• log x = 0.6, x = 3.981

• log x = 0.7, x = 5.012

• log x = 0.8, x = 6.310

• log x = 0.9, x = 7.943

• log x = 1.0, x = 10.00

• x = 10-4, log x = -4

• x = 10-3, log x = -3

• x = 10-2, log x = -2

• x = 10-1, log x = -1

• x = 100, log x = 0

• x = 101, log x = 1

• x = 102, log x = 2

• x = 103, log x = 3

• x = 104, log x = 4

• x = 105, log x = 5

The rules for combining and manipulating logarithms are given below:

• Logarithm of a product, where A and B are numbers, the rule is log (A × B) = logA + logB

• Logarithm of a ratiob, where A and B are numbers, the rule is log (A/B) = logA - logB

• Logarithm of a power, where n is the power of A, the rule is log (An) = n × logA

Logarithms provide a simple alternative to the multiplication and division of large and small numbers. In the days before electronic calculators, this was essentially the only way to quickly manipulate numbers. To multiply two or more numbers in scientific notation, take the logarithms of the numbers and add them. To divide numbers in scientific notation, take the logarithms of the numbers and subtract them. To square a number in scientific notation, take its logarithm and multiply by two. To raise a number in scientific notation to the 1/4 power, take its logarithm and multiply by 1/4. This simple and versatile system has been used for over 300 years.

There is a second kind of logarithm, called a natural logarithm. Natural logarithms provide the easiest way to describe biological and natural phenomena, where an increase occurs by doubling and doubling again. For example, cancer cells or animal population with no resource limitations grows rapidly in a way that is described by an exponential progression. A natural logarithm is also called a logarithm to the base e, where e is the transcendental number 2.718� Instead of x = 10y and y = log10x, we have x = ey and y = logex. Common logs and natural logs are not fundamentally different systems; they are related by a simple numerical factor. Common logarithms are typically used in astronomy and physics. Natural logarithms are typically used in biology and chemistry.