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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?

1 Answer

1 vote

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.

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