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?

Question

asked May 7, 2020 139k views

1 Answer

Analia · Answer 1 · 2020-05-13T16:05:01+0000

Answer:

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.