To solve the equation [tex]\( 2 \log_2 x - \log_2 (2x) = 3 \)[/tex], follow these steps:
1. Express the equation:
[tex]\[
2 \log_2 x - \log_2 (2x) = 3
\][/tex]
2. Use logarithm properties: Recall that [tex]\( \log_b (mn) = \log_b m + \log_b n \)[/tex]. Applying this property to [tex]\( \log_2 (2x) \)[/tex]:
[tex]\[
\log_2 (2x) = \log_2 2 + \log_2 x
\][/tex]
3. Simplify the equation: Substitute the expansion [tex]\( \log_2 2 + \log_2 x \)[/tex] into the original equation:
[tex]\[
2 \log_2 x - (\log_2 2 + \log_2 x) = 3
\][/tex]
4. Combine like terms: Since [tex]\( \log_2 2 = 1 \)[/tex]:
[tex]\[
2 \log_2 x - 1 - \log_2 x = 3
\][/tex]
[tex]\[
\log_2 x - 1 = 3
\][/tex]
5. Isolate [tex]\(\log_2 x\)[/tex]:
[tex]\[
\log_2 x = 4
\][/tex]
6. Solve for [tex]\(x\)[/tex]: Recall that [tex]\( \log_2 x = 4 \)[/tex] implies [tex]\( x = 2^4 \)[/tex]:
[tex]\[
x = 16
\][/tex]
Therefore, the solution to the equation [tex]\( 2 \log _2 x - \log _2(2 x) = 3 \)[/tex] is [tex]\( x = 16 \)[/tex], which corresponds to option C.
Thus, the correct answer is:
[tex]\[ \boxed{x = 16} \][/tex]
or in the provided choices,
[tex]\[ \text{C. } x = 16 \][/tex]
