Logs in Different Bases

Date: 6/11/96 at 21:40:53
From: Anonymous
Subject: Logs in different bases

Suppose a and b are positive numbers for which

log (a) = log (b) = log  (a+2b).  Find b/a.
   9        15         25 

Thanks for your help.
  - Michael

Date: 6/12/96 at 5:53:9
From: Doctor Pete
Subject: Re: log problem

Here are some hints:

Let n = log (a) = log (b) = log  (a+2b).
           9         15        25

Then 9^n = a, 15^n = b, and 25^n = a+2b.

Factoring the bases, we see that 

3^(2n) = a, 3^n * 5^n = b, and 5^(2n) = a+2b.  

Look at these relations carefully, and you will find that you can 
express two of these in terms of the third, thereby eliminating n.  
Once you do this, you have a relation in a and b which allows you to 
find b/a.

Another thing to keep in mind: note b/a must be positive; this should
eliminate any extraneous solutions you may obtain.

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