Answer :
To solve the equation [tex]\(2 \log_2 x - \log_2(2x) = 3\)[/tex], let's break it down step by step.
1. Express the logarithm: The equation [tex]\(2 \log_2 x - \log_2(2x) = 3\)[/tex] involves logarithms, so let's simplify the expression on the left-hand side using properties of logarithms.
2. Logarithm properties: Recall the logarithm properties:
- [tex]\(\log_b(mn) = \log_b m + \log_b n\)[/tex]
- [tex]\(\log_b(m/n) = \log_b m - \log_b n\)[/tex]
We need to simplify [tex]\(\log_2(2x)\)[/tex]. Using the property that [tex]\(\log_b(mn) = \log_b m + \log_b n\)[/tex], we can write:
[tex]\[ \log_2(2x) = \log_2 2 + \log_2 x \][/tex]
Since [tex]\(\log_2 2 = 1\)[/tex], we can substitute:
[tex]\[ \log_2(2x) = 1 + \log_2 x \][/tex]
3. Substitute and simplify: Substitute [tex]\(\log_2(2x)\)[/tex] in the original equation.
[tex]\[ 2 \log_2 x - (1 + \log_2 x) = 3 \][/tex]
Simplify the left-hand side:
[tex]\[ 2 \log_2 x - 1 - \log_2 x = 3 \][/tex]
Combine the logarithms:
[tex]\[ \log_2 x - 1 = 3 \][/tex]
4. Isolate the logarithm:
[tex]\[ \log_2 x = 4 \][/tex]
5. Solve for [tex]\(x\)[/tex]: Recall the definition of a logarithm. If [tex]\(\log_2 x = 4\)[/tex], then:
[tex]\[ x = 2^4 \][/tex]
Calculate the power:
[tex]\[ x = 16 \][/tex]
So, the correct answer is [tex]\(C. x = 16\)[/tex].
1. Express the logarithm: The equation [tex]\(2 \log_2 x - \log_2(2x) = 3\)[/tex] involves logarithms, so let's simplify the expression on the left-hand side using properties of logarithms.
2. Logarithm properties: Recall the logarithm properties:
- [tex]\(\log_b(mn) = \log_b m + \log_b n\)[/tex]
- [tex]\(\log_b(m/n) = \log_b m - \log_b n\)[/tex]
We need to simplify [tex]\(\log_2(2x)\)[/tex]. Using the property that [tex]\(\log_b(mn) = \log_b m + \log_b n\)[/tex], we can write:
[tex]\[ \log_2(2x) = \log_2 2 + \log_2 x \][/tex]
Since [tex]\(\log_2 2 = 1\)[/tex], we can substitute:
[tex]\[ \log_2(2x) = 1 + \log_2 x \][/tex]
3. Substitute and simplify: Substitute [tex]\(\log_2(2x)\)[/tex] in the original equation.
[tex]\[ 2 \log_2 x - (1 + \log_2 x) = 3 \][/tex]
Simplify the left-hand side:
[tex]\[ 2 \log_2 x - 1 - \log_2 x = 3 \][/tex]
Combine the logarithms:
[tex]\[ \log_2 x - 1 = 3 \][/tex]
4. Isolate the logarithm:
[tex]\[ \log_2 x = 4 \][/tex]
5. Solve for [tex]\(x\)[/tex]: Recall the definition of a logarithm. If [tex]\(\log_2 x = 4\)[/tex], then:
[tex]\[ x = 2^4 \][/tex]
Calculate the power:
[tex]\[ x = 16 \][/tex]
So, the correct answer is [tex]\(C. x = 16\)[/tex].
