To find an expression equivalent to [tex]\(\ln \left(\frac{2e}{x}\right)\)[/tex], we can use the properties of logarithms to simplify it step-by-step.
Step 1: Use the quotient rule for logarithms.
The quotient rule states that [tex]\(\ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex]. Applying this rule to our expression:
[tex]\[
\ln \left(\frac{2e}{x}\right) = \ln(2e) - \ln(x)
\][/tex]
Step 2: Use the product rule for logarithms.
The product rule states that [tex]\(\ln(ab) = \ln(a) + \ln(b)\)[/tex]. Applying this to [tex]\(\ln(2e)\)[/tex]:
[tex]\[
\ln(2e) = \ln(2) + \ln(e)
\][/tex]
Step 3: Simplify [tex]\(\ln(e)\)[/tex].
By definition of the natural logarithm, [tex]\(\ln(e) = 1\)[/tex]. Therefore,
[tex]\[
\ln(2e) = \ln(2) + 1
\][/tex]
Step 4: Substitute back into the expression.
Replacing [tex]\(\ln(2e)\)[/tex] in the original equation:
[tex]\[
\ln \left(\frac{2e}{x}\right) = \ln(2e) - \ln(x) = \ln(2) + 1 - \ln(x)
\][/tex]
Thus, the expression [tex]\(\ln \left(\frac{2e}{x}\right)\)[/tex] simplifies to:
[tex]\[
\ln(2) + 1 - \ln(x)
\][/tex]
Step 5: Identify the equivalent expression from the given choices.
Looking at the provided options:
- A. [tex]\(1 + \ln 2 - \ln x\)[/tex]
- B. [tex]\(\ln 2 + \ln x\)[/tex]
- C. [tex]\(\ln 1 + \ln 2 - \ln x\)[/tex]
- D. [tex]\(\ln 2 - \ln x\)[/tex]
We can see that option A is [tex]\(1 + \ln 2 - \ln x\)[/tex], which matches the simplified expression [tex]\(\ln(2) + 1 - \ln(x)\)[/tex].
Therefore, the correct choice is:
[tex]\[
\boxed{1}
\][/tex]
