Question 4281
<font face = "courier new">Prove that:
(log<sub>a</sub>x·log<sub>b</sub>x)/(log<sub>a</sub>x + log<sub>b</sub>x) =log<sub>ab</sub>x
`
We must assume that a, b, and x are all positive 
and different from 1, otherwise the above is not defined.
`
We need two rules of logarithms
(1) log<sub>m</sub>n = 1/log<sub>n</sub>m, when m and n are positive but neither is 1
(2) log<sub>m</sub>p + log<sub>m</sub>q = log<sub>m</sub>(pq), when m,p,q are positive and m is not 1
`
Use (1) to replace every term in left hand side

<PRE><font size = 3>
      1         1
   ------- · -------
    log<sub>x</sub>a     log<sub>x</sub>b
-----------------------
    1          1
 ------- + --------
  log<sub>x</sub>a      log<sub>x</sub>b


Multiply top and bottom by log<sub>x</sub>a·log<sub>x</sub>b


        1
--------------------
 log<sub>x</sub>b + log<sub>x</sub>a
 
      1
------------
  log<sub>x</sub>(ba)
 


` use (1)

       1
 ------------
       1
   --------
    log<sub>ba</sub>x
 

     log<sub>ba</sub>x

     log<sub>ab</sub>x

</PRE>

Edwin