Characteristic and Mantissa of a Common LogarithmDate: 04/24/2003 at 09:11:37 From: Leslie Maruri Subject: Characteristic and mantissa of a common logarithm How do you find the characteristic and mantissa of a negative logarithm? I am a high school teacher and different sources give different ways of computing the character: for example, log(0.05) Character of -3 or -2 Some say to -2.9957 + 10 -10 should you only add the integers and leave the decimal .9957 as the mantissa? If by definition the character is the power of 0.05 = 5x10^-2 why is the characteristic -3 rather than -2? Date: 04/24/2003 at 09:29:40 From: Doctor Jerry Subject: Re: Characteristic and mantissa of a common logarithm Hi Leslie, I'll assume that the base of the logarithms is 10. log(0.05) = log(5*10^{-2}) = log(5) + log(10^{-2}) = log(5) +(-2)* log(10) = log(5) -2*1 = log(5) - 2. Now, log(5) = 0.69897... and so log(0.05) = 0.69897 - 2 = -1.30103 Some like to write this as log(0.05) = 0.69897 - 2 = 0.69897 +8 - 10 = 8.69897 - 10. In olden days (I can remember doing this in 1947, before the advent of electronic calculators in the 1950s) there were log tables in the backs of some math books. To limit the size of the tables, they were just the logs of numbers x for which 1 <= x < 10. When looking up the log of any positive number x one would write x = d * 10^{n}, where 1 <= d < 10 and n a positive or negative integer. log(x) = log(d) + log(10^n) = log(d) + n. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 04/24/2003 at 09:34:49 From: Doctor Mitteldorf Subject: Re: Characteristic and mantissa of a common logarithm Dear Leslie, First, let's distinguish between common logs and natural logs. The concepts of characteristic and mantissa are really useful only for common logs, to the base 10. You quote log(.05) = -2.9957. This is the natural log, to the base e. So let's talk about common (base 10) logs, and see why mantissa and characteristic are useful concepts. log(5000) = 3.69897 log(500) = 2.69897 log(50) = 1.69897 log(5) = 0.69897 log(.5) = -.30103 = -1 + 0.69897 log(.05) = -1.30103 = -2 + 0.69897 The first thing to notice is that all the numbers we're taking logs of are 5*10^n, for different integers n, and that the mantissa of each of these is the same. Why is that? This in itself is a lesson of discovery that you can pursue on your own, then encourage your students to recreate the experience for themselves. Now imagine the world the way it was in the old days, before calculators let us compute logs with the touch of a button. There were big books full of log tables. In constructing the tables, it didn't make sense to list the log of every number from 1 to a billion. Instead, the table used a shortcut: List the numbers from 1.000 to 9.999 with their logarithms. Then ask the person reading the table to adjust. For example, to find the log of the number 2345, he would be asked to look up the log for 2.345, and then add 3 to it, because 2345 is just 2.345 times 10^3. In order to make this system work for numbers less than 1, (with negative logs), the idea was introduced that -1.30103 should be treated not as -1 added to -0.30103, but instead -2 added to 0.69897. This way of doing the problem maintains our correspondence betweeen notation for the decimal numbers and the mantissa from the table. The simple procedure would then be: To find the log of 0.05, first look up the log of 5, then subtract 2 from that number (because 0.05 is just 5 times 10^-2). - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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