What is the true solution to the logarithmic equation?

[tex]\[ \log_2\left[\log_2(\sqrt{4x})\right]=1 \][/tex]

A. [tex]\( x = -4 \)[/tex]
B. [tex]\( x = 0 \)[/tex]
C. [tex]\( x = 2 \)[/tex]
D. [tex]\( x = 4 \)[/tex]



Answer :

Sure, let's solve the equation step-by-step to determine which value of [tex]\( x \)[/tex] is the true solution. The equation given is:

[tex]\[ \log_2\left( \log_2(\sqrt{4x}) \right) = 1 \][/tex]

Let's go through the steps to solve this equation:

1. Isolate the outer logarithm:

The equation is:
[tex]\[ \log_2\left( \log_2(\sqrt{4x}) \right) = 1 \][/tex]

Since the base of the log is 2, we can convert the equation to its exponential form:
[tex]\[ \log_2(\sqrt{4x}) = 2^1 \][/tex]
[tex]\[ \log_2(\sqrt{4x}) = 2 \][/tex]

2. Isolate the inner logarithm:

The new equation is:
[tex]\[ \log_2(\sqrt{4x}) = 2 \][/tex]

We convert this to its exponential form as well:
[tex]\[ \sqrt{4x} = 2^2 \][/tex]
[tex]\[ \sqrt{4x} = 4 \][/tex]

3. Remove the square root:

To eliminate the square root, square both sides of the equation:
[tex]\[ 4x = 4^2 \][/tex]
[tex]\[ 4x = 16 \][/tex]

4. Solve for [tex]\( x \)[/tex]:

Divide both sides of the equation by 4:
[tex]\[ x = \frac{16}{4} \][/tex]
[tex]\[ x = 4 \][/tex]

Through this process, we can see that the true solution to the equation is:

[tex]\[ x = 4 \][/tex]

So, out of the given options, the true solution is:

[tex]\[ x = 4 \][/tex]