'Log' Button

Date: 01/29/98 at 11:00:58
From: Brad Edey
Subject: The 'log' button

In math we have been using the 'log' button on our calculators to 
solve problems that involve compound interest, etc., and our teacher 
says we'll learn what the 'log' button does next year. But we want to 
know now!  The teacher says it has something to do with the opposite 
of exponents, but I still have no clue.

Date: 01/29/98 at 15:23:30
From: Doctor Sam
Subject: Re: The 'log' button

Well, logarithms can be a big topic and if you are interested you 
might try reading about them in another math book. But for a quick 
introduction, here goes:

First, the word logarithm is really a synonym for the word exponent or 
power.

Second, there are lots of different kinds of logarithms depending upon 
the base you are using.

The LOG key on your calculator is probably "base 10 logarithm".  Try 
this: make a table of values of different powers of 10, like this:

        power x:   -2   -1  0   1    2     3  ...
           10^x  0.01  0.1  1  10  100  1000  ...

Ordinarily we read these as  "10 to the power 2 equals 100" or "10 to 
the power 3 equals 1000."

This is fine when you need to calculate a power of ten, but what 
if you know the answer and need to find the exponent?  To solve 
10^x = 10000, for example, you have to think "ten to what power is 
ten thousand?" That's not too hard. But what if the problem is 
10^x = 25?  

LOGARITHMS were invented (back in the sixteenth century I think) to 
answer these kinds of questions.

LOG (25) means "the power of 10 that produces 25"   and
LOG(1000) = 4 because "the power of 10 that produces 1000 is 4."

You can make a little table of logarithms just by switching the two 
rows of my table above:

     10^x    0.01   0.1   1   10   100   1000  ...
  power x:     -2    -1   0    1     2      3  ...

only now the second row is called the logarithm. So here it is one 
more time:

        n    0.01   0.1   1   10   100   1000 ...
    LOG(n):    -2    -1   0    1     2      3    ...

Back in the sixteen hundreds (in fact even back in the dark ages when 
I went to school in the 1960's) we didn't have pocket calculators with 
LOG keys. Companies published books of tables, page after page of the 
powers of 10 that would give you almost any number you wanted. It's a 
lot easier to press a key!

I do want to mention that the little table I used gives powers of 10, 
and so these are called base ten logarithms (or COMMON LOGARITHMS).  
But almost any number can be used as the base. If you are interested 
in solving problems with powers of 2 (175 = 2^x) it would be helpful 
to have a base two logarithm key on your calculator. You probably 
don't have one, but computer scientists who use powers of two a lot 
do.  

I hope that helps...

-Doctor Sam,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/