Simplifying the equation we have

Dividing through by , the common factor we have

changing the base we have

Hence  or

Let  or

Factorizing and solving we have
 or
But
If
Also if

To show his we simply change the base of the logarithm thus;

Considering the given simultaneous equations

 --------(1)
---------------(2)
From equation
-------(3)
From equation (2)
 ----(4)
Substituting  in  we have

From equation (3) 

We still apply change of base thus

Let 
Factorizing we have and solving we have


 or
But
Solog  or
 or

Changing the base and simplifying we have.

Let 
Factorizing and solving we have
 or

But
Hence
or
But from the question  or  and  are unequal
Hence  is the right relation.

In this problem we change the base of the logarithm thus

Multiplying through by  and rearranging we have

Let
Factorising we and solving we have

 or
But
If
If

a)

To solve this question we may proceed thus Let
Then

 mog
This problem could also be solved by changing base 4 to base 2 thus:
Surely you can show that  )
b)

We can solve this problem by changing the logarithm  to base 7 or  to base 5

Multiplying through by  we have  or

Factorizing and solving we have
 or
If  and
If

Multiplying through by  we have

Reordering the equation we have

Factorizing we have

Solving we have
 or .
If
If  or

We can rewrite the equation as

Let  then the equation becomes

So
Hence  or
But , therefore  or
If  then  or  (recall the laws of logaritims)
if  then  is undefined.

Multiply through by  we have

Reordering the equation we have

Factorizing we have

Solving we have  or
If  then
If  then  or