A:

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

A:

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

A:

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) 

A:

We still apply change of base thus

Let 
Factorizing we have and solving we have


 or
But
Solog  or
 or

A:

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.

A:

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:

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

A:

Multiplying through by  we have  or

Factorizing and solving we have
 or
If  and
If

A:

Multiplying through by  we have

Reordering the equation we have

Factorizing we have

Solving we have
 or .
If
If  or

A:

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.

A:

Multiply through by  we have

Reordering the equation we have

Factorizing we have

Solving we have  or
If  then
If  then  or

A:

Multiplying through by , we have
So
Factorizing we have
Solving we have  of
If
sine
 hn sha inle
1e

A:

Let .
Factorizing we have

So  or

Also  or  is not real

A:

.
We can express the given equation thus.
 
Now let
Factorizing we have
Hence  or
,
If
If  then  i.e.  is undefined

A:

Simplifying the equation we have

Let  then
Dividing through by 2 the common factor to the terms,

Factorizing we have
Hence  or
But , so  or
Therefore  or  i.e.  is undefined.

A:

We rewrite the equation thus:

Let  then
Factorizing we have

 or
But
Hence  or
If , then
Also if  (  is undefined or  is notareal number)

A:

We express the equation as

Let
Factorising we have
 or
 or
But , so  or
Considering
We apply logarithms thus


If  then,  which means no real number satishes the equation.

A:

, then
Factorising, we have

 or
But  or   or
By comparison of powers we have  or

A:

(i)

(ii)
(iii)

A:

a)

From the table above the last two columns are identical. Hence ∼p∧∼q and ∼(p∨q) are logically equivalent
b)

The last two columns of the table above are identical. Hence ∼(p∧q) and ∼p∨∼q are logically equivalent