If  then

Hence 

Applying change of base we have

Applying the laws of logarithms we have .

The LHS and RHS are logarithms to the same base; hence we compare the functions thus

We divide through by y in order to get an equation in terms of what we need.
Let
 or
Hence  or

We factorize directly in terms of natural logarithms thus
 or

………(1)

……….(2  )

Using equation (1) we change  to base 2 and simplifying the equation we have

Changing from logarithmic to index form we have

……..(3)

From equation (2)
Changing from logarithmic form to index form we have
 …….(4)
Substituting (4) in (3) we have

Factoring out 2 from the bracket we have

Multiplying through by y and dividing through by  we have

Factorizing the quadratic and solving we have

We can prove this by applying logarithms thus

Using the laws of logarithms we have

Grouping like terms we have

a)
Changing to base 2 we have
Rearranging the equation we have
Factorizing directly in logarithmic form we have
Solving we have
or
b]
changing to base 3 and simplifying

Factorizing and solving we have
Hence

or

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