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]
