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Select the correct answer.

What is the solution to this equation?
[tex]\[
2 \log_2 x - \log_2 (2x) = 3
\][/tex]

A. [tex]\(x = 18\)[/tex]
B. [tex]\(x = 8\)[/tex]
C. [tex]\(x = 16\)[/tex]
D. [tex]\(x = 6\)[/tex]

Sagot :

GinnyAnsweridn
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].
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