To determine which logarithmic expressions are equivalent to \(\ln(\sqrt{xy}) + \ln\left(\frac{x}{e}\right)\), we can start simplifying the given expression:
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The expression \(\ln(\sqrt{xy})\) can be rewritten using properties of logarithms: \[ \ln(\sqrt{xy}) = \ln((xy)^{1/2}) = \frac{1}{2} \ln(xy) = \frac{1}{2} (\ln x + \ln y) \]
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The expression \(\ln\left(\frac{x}{e}\right)\) can also be simplified: \[ \ln\left(\frac{x}{e}\right) = \ln(x) - \ln(e) = \ln(x) - 1 \]
Combining both parts gives us: \[ \ln(\sqrt{xy}) + \ln\left(\frac{x}{e}\right) = \frac{1}{2} (\ln x + \ln y) + (\ln x - 1) \]
- We can simplify this further: \[ = \frac{1}{2} \ln x + \frac{1}{2} \ln y + \ln x - 1 \] \[ = \left(\frac{1}{2} \ln x + \ln x\right) + \frac{1}{2} \ln y - 1 \] \[ = \left(\frac{3}{2} \ln x\right) + \frac{1}{2} \ln y - 1 \]
Now, we will compare the simplified expression with the given options:
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Option A: \(\ln\left(\frac{\sqrt{x^3y}}{e}\right)\)
Simplifying this: \[ = \ln\left(\frac{x^{3/2} \sqrt{y}}{e}\right) = \ln(x^{3/2}) + \ln(\sqrt{y}) - \ln(e) \] \[ = \frac{3}{2} \ln x + \frac{1}{2} \ln y - 1 \] This is equivalent.
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Option B: \(\ln\left(\frac{x^2y}{2e}\right)\)
Simplifying: \[ = \ln\left(\frac{x^2y}{2}\right) - \ln(e) = \ln(x^2) + \ln(y) - \ln(2) - 1 \] \[ = 2\ln x + \ln y - \ln 2 - 1 \] This is not equivalent.
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Option C: \(\frac{1}{2} \ln(x^2y) - e\)
This expression is not equal to our simplified version.
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Option D: \(\ln(x\sqrt{xy}) - 1\)
Simplifying: \[ = \ln(x \cdot (xy)^{1/2}) - 1 = \ln(x^{3/2}y^{1/2}) - 1 = \frac{3}{2} \ln x + \frac{1}{2} \ln y - 1 \] This is equivalent.
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Option E: \(\frac{3}{2} \ln x + \frac{1}{2} \ln y - 1\)
This is directly the equivalent expression we simplified to.
Conclusion:
The equivalent logarithmic expressions are:
- Option A: \(\ln\left(\frac{\sqrt{x^3y}}{e}\right)\)
- Option D: \(\ln(x\sqrt{xy}) - 1\)
- Option E: \(\frac{3}{2}\ln x + \frac{1}{2}\ln y - 1\)
So the correct responses are A, D, and E.