The Special Properties of Logarithms

Converting Multiplication into Addition

• Adding the logarithms of two numbers gives the logarithm of the product of the two numbers. This means that if you can convert easily between numbers and their logarithms you can perform multiplications by calculating addition only, making calculations a lot easier. Slide rules had two scales that were logarithmic and adding two lengths together gives the product of the numbers on the logarithmic scale. For example, 0.3010 is the log of 2 and 0.4771 is the log of 3. and the log of (2 X 3) is 0.3010 + 0.4771. Reducing multiplication to addition becomes even more important when you are multiplying 10- or 20-digit numbers -- as astronomers and physicist do.

Converting Division into Subtraction

• To divide A by B you can use the relationship Log (A/B) = Log(A) - Log(B). Logarithms reduce division to subtraction -- a much easier calculation. Notice that the multiples of 10 of a number is denoted by the number to the left of the decimal in the logarithm. So, 0.3010 is the logarithm of 2, but 1.3010 is the logarithm of 20 and 2.3010 is the logarithm of 200. The logarithm of 300/20 would be 2.4771 - 1.3010 = 1.1070 which is clearly a number between 10 and 100. In fact, it's the logarithm of 15.

Converting Exponentiation into Multiplication

• Logarithms also reduce the calculation of exponents to multiplication. A to the B power can be converted into a multiplication problem by the relationship Log(A^B) = B X Log (A). To find the 5th power of 2, multiply the logarithm of two by five to get 5 X 0.3010 = 1.5050, which is the logarithm of 32.

Beyond Computation

• Logarithms were developed to simplify calculations, but if that was their only use, logarithms would be as obsolete as the slide rule. Logarithms are the inverse of exponentiation and you can see this in the way mathematicians use the phrase logarithmic as the inverse of exponential. For example, we say that germ growth is exponential but radioactive decay is logarithmic. Logarithms have remained a valuable concept because they explain natural processes -- for example, the area under the simple curve y = 1/x between two values of x is the difference of the logarithms of the two points.