Exponential Equations

Date: 01/19/2003 at 18:33:06
From: James Boulter
Subject: Exponent Equations

Is there a mathmatical way to solve 2^x + 3^x = 13 ? Obviously, x = 2, 
but is there an algebraic way of calculating the answer, or do we just 
have to use common sense?

Date: 01/20/2003 at 16:58:10
From: Doctor Ian
Subject: Re: Exponent Equations

Hi James,

Suppose we change the equation a little:

  2^x + 3^x = 14

Now the only way to find the answer is to use iteration - that is,
guess, check each guess, and use the check to make a new guess:

     x    2^x + 3^x  Result
  ------  ---------  ---------
  2       13         Too small
  2.1     13.7       Too small
  2.2     15.4       Too big
  2.15    14.56      Too big
  2.125   ...        ...

If the answer is rational, we'll find out in a finite number of steps.
Otherwise, we just have to quit when we think the answer is good
enough. 

Now, suppose we go back to the original equation, 

  2^x + 3^x = 13

and start with a guess, like 1, and then flail around a little:

     x    2^x + 3^x  Result
  ------  ---------  ---------
  1       5          Too small
  4       97         Too big
  2.5     ...        ...

Eventually, we'll narrow the value of x down to exactly 2. Just how
long that will take will depend on how smart we are about refining our
guesses. 

Now, this is also 'common sense', isn't it?  That is, we know that the
function increases when x > 1, so if we bracket the number on the
right side, we'll eventually find the solution - or get as close to
the solution as we wish. All you've done is to short-circuit the 
process by making a really good initial guess.  

But there is nothing 'non-mathematical' about making good guesses! 
In fact, there are many cases where we have no choice but to make
guesses. See our FAQ on 'Segments of Circles', 

   http://mathforum.org/dr.math/faq/faq.circle.segment.html 
  
for several examples of such cases. 

Does this help? 

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