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Composite Functions Using Logarithms

Date: 3/10/96 at 21:33:41
From: Anonymous
Subject: Precal

I have the following precal question and am having a bit
of difficulty with it.  Any help (even an idea about how to
approach it) would be helpful.  Thanks so much.

Suppose f and g are functions defined by f(x) = x+2 and 
g(x) = x. Find all x > -2  for which 

                g(x) * logbase3 f(x)
               3                       = f(x)

In case you can't read that, it says 

3 to the exponent (g(x) * the log f(x) base3) equals f(x).  


Date: 3/21/96 at 13:32:36
From: Doctor Aaron
Subject: Re: Precal


Good question.  

A good place to start is to see that you have an expression that
is close to the form:

  logbasea (f(x))
a                 .  

If this doesn't look useful, think about what the log function 
does. It is essentially the inverse of the exponential 
function (provided the exponential and the log have the same 

Composing a function with its inverse gives us what is called the 
identity function - it gives us back whatever we give it to 
operate on. That is, if we raise a number to a power, and then 
take the log (of the appropriate base) we get back the power to 
which we raised our number.  

Similarly, if we take the log of a number, then raise the base to 
the power (log(base)), we get our number back.  

so then   

 logbasea (f(x))
a                 = f(x).  

You may have noticed a problem.  You have g(x) in front of the 
log, so we can't simplify as nicely as we would like to.  We can 
change this by using the relation that a*log(b) = log(b^a) where 
^ is an exponent.

This relation makes sense, because the log just asks us how many 
times we have to multiply a number by itself to get another 
number. (Actually we can't multiply something by itself a 
fractional number of times, but log is just the extension of the 
above definition to the real numbers)

Well, if we have to muliply (base) by itself n times to get b, 
we'll have to multiply (base) by itself a*n times to get b^a.

Now rewrite  g(x) * logbase3 f(x) as logbase3 (f(x)^g(x)).

You should be able to use the above ideas to make some headway.
If you run into some more trouble, consult a teacher or write 

-Doctor Aaron,  The Math Forum

Associated Topics:
High School Functions
High School Logs

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