What Does x Equal?

Date: 01/16/98 at 18:15:44
From: Amanda Hoffman
Subject: Algebra 2/Trigonmetry

Dr. Math,

We have been doing problems with logarithms. Can you help me with this 
problem?
           
   32 times 4 to the 2x power equals 12. 

I have to know what x equals.
                
Also, what is a logarithm?

Amanda

Date: 01/21/98 at 22:59:19
From: Doctor Wolf
Subject: Re: Algebra 2/Trigonmetry

Hi Amanda,

To answer your last question first ... a logarithm IS an exponent.
There is an excellent article in the ARCHIVES entitled "Hints About
Logs."  It would be good for you to review this short document on logs 
and their properties often as you progress in your class:
  
   http://mathforum.org/dr.math/problems/loghints.html   

Being able to convert logarithmic equations to exponential equations 
and back is essential to solving log problems.

Back to the problem you presented: 

   32*4^(2x) = 12

Dividing both sides of this equation by 32 we have

   4^(2x) = 12/32

Let's do a little work on the left side of this equation first.           
  
   4^(2x) = (4^2)^x = 16^x    using the multiplication property of
                              exponents and the fact that 
                              4^2(squared) equals 16.
Now for the right side: 
 
    12/32 = 6/16      I reduced the fraction, but wanted 16 in the
                      denominator for a reason which will become
                      clear in the next few steps.
Therefore:

     16^x = 6/16      using the new left and right sides from above.

But this means that x must be the log of 6/16 to the base 16, by the 
definition of a logarithm. We can go just a bit further:

Since 
            log (a/b) = (log a) - (log b) (division prop. of logs)
  log (6/16)(base 16) = log 6 (base 16) - log 16 (base 16)
                      = log 6 (base 16) - 1    Why?

Finally:            x = log 6 (base 16) - 1

I hope this helped. It's not really what I would call a trivial
problem, and may have various forms for an answer.

Don't hesitate to get back to me if you have further questions.

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