Question

Which of the following logarithmic expressions are equivalent to natural log start root x y end root plus natural log left parenthesis start fraction x over e end fraction right parenthesis? Select all that apply.
(2 points)
Responses

natural log left parenthesis start fraction start root x cubed y over e end fraction right parenthesis
Image with alt text: natural log left parenthesis start fraction start root x cubed y over e end fraction right parenthesis

natural log left parenthesis start fraction x squared y over 2 e end fraction right parenthesis
Image with alt text: natural log left parenthesis start fraction x squared y over 2 e end fraction right parenthesis

one half natural log x squared y minus e
Image with alt text: one half natural log x squared y minus e

natural log left parenthesis x start root x y end root right parenthesis minus 1
Image with alt text: natural log left parenthesis x start root x y end root right parenthesis minus 1

Start Fraction 3 over 2 End Fraction natural log x plus Start Fraction 1 over 2 End Fraction natural log y minus 1

Answer 1

To determine which logarithmic expressions are equivalent to \(\ln(\sqrt{xy}) + \ln\left(\frac{x}{e}\right)\), we can start simplifying the given expression:

1. The expression \(\ln(\sqrt{xy})\) can be rewritten using properties of logarithms:
\[
\ln(\sqrt{xy}) = \ln((xy)^{1/2}) = \frac{1}{2} \ln(xy) = \frac{1}{2} (\ln x + \ln y)
\]

2. 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:
\[
\ln(\sqrt{xy}) + \ln\left(\frac{x}{e}\right) = \frac{1}{2} (\ln x + \ln y) + (\ln x - 1)
\]

3. 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:

1. **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.

2. **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.

3. **Option C**: \(\frac{1}{2} \ln(x^2y) - e\)

This expression is not equal to our simplified version.

4. **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.

5. **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.