Does Repetitive Division Correspond to a Logarithm?

Date: 05/26/2008 at 19:20:08
From: Amir
Subject: How to determine from division if a function is logarithmic

Hello,

I was in a class recently and it's been a few years since I studied
exponents and logarithms but my interest in them never ceases!  I
understand that logs and exponents are inverse functions and how they
correlate in that sense.  I also understand that when you have a
number and you continuously multiply it by itself this is considered
exponential and I know what the graph of that looks like, e.g., 
f(x) = 2^x. 

Where I got a bit confused was when the teacher mentioned, "Well if we
continuously divide this number then what do we have?  A logarithmic 
function."

So can we perceive logs in another way as this: since exponents are
just the multiplication of numbers, and since logs are the inverse
function, then the division of a number is similar to a logarithmic
function.  Basically, can we correlate that somehow?

I haven't really been able to prove much of this idea that I have that
correlates multiplication/division to exponents/logs.  I've searched
online but to no avail.

Date: 05/27/2008 at 09:12:54
From: Doctor Ian
Subject: Re: How to determine from division if a function is logarithmic

Hi Amir,

This is a good question.  Thanks for asking it.  I've never really
thought about it that way!  Let's see what he might mean by it.

Let's leave aside for the moment the continuous nature of both 
exponents and logarithms, e.g., we can raise any real base to any real
exponent, e.g., 

  pi^sqrt(2) = 5.0474969 (approximately)

and just focus on integers. 

We can look at something like 

  3^4

and interpret that as multiply four copies of 3

  = 3 * 3 * 3 * 3

  = 81

Now, the first thing to note it that there are TWO ways to invert this
process.  If we know the exponent, we use a root to find the original
base:

   4___
  \/ 81 = 3

And if we know the base, we can use the logarithm to find the 
exponent:

  log_3(81) = 4

We need two different inverses, because the operators aren't 'equal
partners', as they are in multiplication.  

The root is answering the question:

  If we multiply four copies of some quantity and get a product
  of 81, what would that quantity be?

The logarithm is answering the question:

  If we multiply some number of copies of 3 and get a product
  of 81, what would that number be? 

Now, suppose we decide we'd like to answer these numerically.  That
is, we'll make a guess, and try it, and use the error to come up with
a better guess.  In the case of the root, we might guess that the root
is 4.  Now we divide by 4, 3 times:

  81     / 4 = 20.25

  20.25  / 4 =  5.065

   5.065 / 4 =  1.266

The result is too small, so we need to try a smaller guess.  How about
3.1?

  81     / 3.1 = 26.129  
  
  26.129 / 3.1 =  8.429

   8.429 / 3.1 =  2.719

Still too small, but getting closer.  (That is, 2.719 is closer to
3.1, than 1.266 was to 4.  I need the initial and final numbers to be
the same.)  Let's try 3:

  81 / 3 = 27  
  
  27 / 3 =  9

   9 / 3 =  3          So 81 = 3 * 3 * 3 * 3 = 3^4

So a root is a succession of divisions, where we know how many times
to divide (one less than the exponent), but not what to divide by (the
base).  Does that make sense?

A logarithm would then deal with the other case:  We know what to
divide by (the base), but not how many times to divide.  Again, we
could do this by trial and error.  Let's guess that 81 = 3^3, so we
divide twice, with the target being to end at 3.

  81 / 3 = 27  
  
  27 / 3 =  9

The result is too high, so we didn't divide enough times.  How about 5?

  81 / 3 = 27  
  
  27 / 3 =  9
 
   9 / 3 =  3

   3 / 3 =  1

Too low.  We divided too many times.  As we know, an exponent of 4--
right in between our two guesses--will be just right.  :^D

As I said, I don't want to get into the complexities involved when we
move from integers to all real numbers.  But do you see the 
relationship between exponents, roots, and logarithms, and how roots
and logarithms can be understood as repeated divisions? 

Does this answer your question? 

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

Date: 07/14/2008 at 22:11:52
From: Amir
Subject: Thank you (How to determine from division if a function is
logarithmic)

Thank you!  You did a great job.