Logarithmic Multiplication

Date: 05/15/2003 at 18:53:24
From: Tess
Subject: Logarithmic Multiplication - Rearranging Bases within Logs

We found that you can rearrange bases in logarithmic multiplication.  
Therefore log(b)x * log(c)y = log(c)x * log(b)y.

Take this problem:

log(2)3 * log(3)64. WIth the change of base theorum we can rewrite it 
as (log 3 / log 2) * (log 64 / log 3), and then solve. If we make the 
base 2 instead of 10, the log(2) 3's cancel out and log(2) 2 becomes 
one, leaving log(2) 64, which equals 6. If you rearrange the bases, 
making it log(3) 3 * log(2) 64, you get 6, the right answer. So we 
have proven that our theorem is true.         

The real question is, then, have we discovered something?  It seems 
to be a useful tool, so why isn't it in a high school textbook's 
logarithm section, and why doesn't our teacher recognize the 
theorem?  I am sure, as it is an easy idea, it has been discovered 
before.  If so, what is the theorem called?  And if not...?

Date: 05/15/2003 at 23:31:52
From: Doctor Peterson
Subject: Re: Logarithmic Multiplication - Rearranging Bases within 
Logs

Hi, Tess.

It's always fun to discover something yourself, whether it's new or 
not. Let's take a look.

You've given an example, but didn't actually prove your general 
statement; so we should just call it a conjecture for now. To prove 
it, we'll have to find a way to get from one side to the other. So 
I'll do just what you did in the example, without doing anything 
special that requires x=c as in the example:

   log_b(x) * log_c(y) = log(x)/log(b) * log(y)/log(c)

                       = [log(x) log(y)] / [log(b) log(c)]

                       = log(x)/log(c) * log(y)/log(b)

                       = log_c(x) * log_b(y)

So it is true in general, and we can now call it a theorem. All that 
your "proof" lacked was to actually apply the method to variables, 
rather than just demonstrate it in a particular case. (What you did 
is like what ancient mathematicians did before algebraic notation was 
invented, where they would show a method or a proof by example rather 
than by a general formula.)

There are two reasons why you haven't seen this in books: first, it 
is just a straightforward application of a basic rule, as you pointed 
out, so it doesn't deserve a special name; second, you probably won't 
see many occasions to apply it in real life, since we don't often take 
logs to different bases in one equation. But it is certainly good to 
have seen it, because this gives you an awareness of the possibilities 
in logarithmic expressions that we often fail to notice, and there 
just might be times when this or something like it will pop up 
somewhere. Logs can be surprisingly flexible, and can be simplified 
when they seem too complicated to do anything with.

It reminds me of a fact that I _am_ aware of, though this likewise 
isn't mentioned too often:

    log_a(b) = 1/log_b(a)

This, too, follows immediately from the change-of-base formula; it can 
even be useful some times. But most often when I find myself using it, 
I just rediscover it in the process of solving a problem. I don't need 
to know it in order to solve problems, but it is fun to see when I 
look back for an easier solution.

Thanks for sharing your observation with us. Keeping trying things, 
and eventually you may well discover something new and important!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/17/2003 at 21:39:39
From: Tess
Subject: Thank you (Logarithmic Multiplication - Rearranging Bases 
within Logs)

Dr. Peterson,

Thank you very much for taking time to answer our question. You 
provided us with a thorough explanation, and even taught us a little 
more about theories, conjectures, and logarithms.

Sincerely,
Tess and the rest of Algebra 2 Honors