Question

     i.       Express  in terms of y, where

   ii.       Hence solve the equation

expressing your answers for x in terms of logarithms where appropriate.

Solution

     i.
 

We are given that , therefore;

Hence,  where .

   ii.
 

We are required to solve the equation;

Let’s substitute ;

As demonstrated in (i),  when , therefore;

It is evident that now we have a quadratic equation to solve.

For a quadratic equation , the expression for solution is;

Therefore;

Now we have two options.

Since we have substituted , therefore;

Taking natural logarithm of both sides in both equations.
Power Rule;
Power Rule;

Hence;