Characteristic and Mantissa of a Common Logarithm

Date: 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/