Section 6: Properties of Logarithms; Solving Exponential Equations

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 8.6 Properties of Logarithms; Solving Exponential Equations 833 Version: Fall 2007 8.6 Properties of Logarithms; Solving Exponential Equations Logarithms were actually discovered and used in ancient times by both Indian and Islamic mathematicians. They were not used widely, though, until the 1600’s, when logarithms simplified the large amounts of hand computations needed in the scientific explorations of the times. In particular, after the invention of the telescope, calcula- tions involving astronomical data became very important, and logarithms became an essential mathematical tool. Indeed, until the invention of the computer and electronic calculator in recent times, hand calculations using logarithms were a staple of every science student’s curriculum. The usefulness of logarithms in calculations is based on the following three important properties, known generally as the properties of logarithms . Properties of Logarithms a) log b ( MN ) = log b ( M ) + log b ( N ) b) log b M N = log b ( M ) log b ( N ) c) log b ( M r ) = r log b ( M ) provided that M, N, b > 0 . The first property says that the “log of a product is the sum of the logs.” The second says that the “log of a quotient is the difference of the logs.” And the third property is sometimes referred to as the “power rule”. Loosely speaking, when taking the log of a power, you can just move the exponent out in front of the log. We won’t go into the details of the computation procedures using properties (a) and (b), since these procedures are no longer necessary after the invention of the calculator. But the idea is that a time-consuming product of two numbers, for example two 10-digit numbers, can be transformed by property (a) into a much simpler addition problem. Similarly, a large and difficult quotient can be transformed by property (b) into a much simpler subtraction problem. Properties (a) and (b) are also the basis for the slide rule, a mechanical computation device that preceded the electronic calculator (very fast and useful, but only accurate to about three digits). Property (c), on the other hand, is still useful for difficult computations. If you try to compute a large power, say 2 100 , on a calculator or computer, you’ll get an error message. That’s because all calculators and computers can only handle numbers and exponents within a certain range. So to compute a large power, it’s necessary to use

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