Question

How many extraneous solutions exist for the logarithmic equation below if it is solved in the most efficient way possible?log_(2)[log_(2)(\sqrt(4x))]=1?

Answer 1

No extraneous solution

Explanation:

We have the logarithmic equation given by,

`\log_(2)[\log_(2)(√(4x))]=1`

i.e.
`\log_(2)(√(4x))=2^(1)`

i.e.
`√(4x)=2^(2)`

i.e.
`√(4x)=4`

i.e.
`4x=4^(2)`

i.e.
`4x=16`

i.e.
`x=4`

So, the solution of the given equation is x=4.

Now, as we domain of square root function is x > 0 and also, the domain of logarithmic function is
`( 0,\infty )`.

Therefore, the domain of the given function is x > 0.

We know that the extraneous solution is the solution which does not belong to the domain.

But as x=4 belongs to the domain x > 0.

Thus, x = 4 is not an extraneous solution.

Hence, this equation does not have any extraneous solution.