SOLUTION: Given that log b (xy^3)=m and log b (x^3y^2)=p, find log b ([square root]xy) in terms of m and p.

Given that log b (xy^3)=m and log b (x^3y^2)=p, find log b ([square root]xy)
in terms of m and p.
 
The three basic principles of logarithms are

(1) logB(MN) = logM + logN

(2) logB(M/N) = logBM + logBN

(3) logBMN = N·logBM

We'll use these three rules on the equations for m and p:

           logb(xy3) = m

Use (1) on left side:

      logbx + logby3 = m  

Use (3) on the second term:

     logbx + 3·logby = m

-------------------------------

Next:

          logb(x3y2) = p 

Use (1)

     logbx3 + logby2 = p       

Use (3) on each of the terms on the left:

   3·logbx + 2·logby = p

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     logbx + 3·logby = m
   3·logbx + 2·logby = p

This is a system of two equations and two unknowns, so we 
solve it by Cramer's rule for logbx and logby

            |m 3|

            |p 2|     2m-3p     2m-3p     -3p+2m      3p-2m
   logbx = ------- = ------- = ------- = --------- = ------- 
            |1 3|      2-9        -7         -7         7
            |3 2|



            |1 m|
            |3 p|     p-3m     p-3m     -3m+p     3m-p
   logby = ------- = ------ = ------ = ------- = ------
            |1 3|     2-9       -7         -7       7
            |3 2|

Now we use the rules on the expression:
        __
   logbÖxy
 
The square root is the same as the 1/2 power, so we write:

   logb(xy)1/2

Use (3)

   (1/2)·logb(xy)

Use (1) on the log

   (1/2)·(logbx + logby)

Now replace logbx by (3p-2m)/7 and logy by (3m-p)/7

   (1/2)·[(3p-2m)/7 + (3m-p)/7]

   (1/2)(1/7)·[(3p-2m) + (3m-p)]

   (1/14)·[3p-2m+3m-p]

   (1/14)·(2p+m)

    (2p+m)/14 

Edwin J