Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

[tex]\[
\ln \sqrt{\frac{x}{y}}
\][/tex]

[tex]\[
\frac{1}{2} \ln x - \frac{1}{2} \ln y
\][/tex]



Answer :

Sure! Let's work through the given logarithmic expressions step-by-step:

### Part 1: Expand \(\ln \sqrt{\frac{x}{y}}\)

First, consider the given expression:
[tex]\[ \ln \sqrt{\frac{x}{y}} \][/tex]

1. Simplify the square root:
Use the property that \(\sqrt{a} = a^{1/2}\):
[tex]\[ \ln \sqrt{\frac{x}{y}} = \ln \left( \left( \frac{x}{y} \right)^{1/2} \right) \][/tex]

2. Power rule of logarithms:
Use the rule \(\ln (a^b) = b \ln (a)\):
[tex]\[ \ln \left( \left( \frac{x}{y} \right)^{1/2} \right) = \frac{1}{2} \ln \left( \frac{x}{y} \right) \][/tex]

3. Quotient rule of logarithms:
Use the rule \(\ln \left( \frac{a}{b} \right) = \ln (a) - \ln (b)\):
[tex]\[ \frac{1}{2} \ln \left( \frac{x}{y} \right) = \frac{1}{2} (\ln (x) - \ln (y)) \][/tex]

Therefore, the expanded form of \(\ln \sqrt{\frac{x}{y}}\) is:
[tex]\[ \ln \sqrt{\frac{x}{y}} = \frac{1}{2} (\ln (x) - \ln (y)) \][/tex]

### Part 2: Simplify \(\frac{1}{-} \ln x - \frac{1}{\ln x}\)

Consider the given expression:
[tex]\[ \frac{1}{- \ln x} - \frac{1}{\ln x} \][/tex]

1. Simplify first term:
The first term can be interpreted as:
[tex]\[ \frac{1}{- \ln x} = -\frac{1}{\ln x} \][/tex]

2. Combine the terms:
Combine the simplified first term with the second term:
[tex]\[ -\frac{1}{\ln x} - \frac{1}{\ln x} = -\frac{1}{\ln x} - \frac{1}{\ln x} \][/tex]

3. Addition of like terms:
[tex]\[ -\frac{1}{\ln x} - \frac{1}{\ln x} = -2 \left( \frac{1}{\ln x} \right) \][/tex]

4. We can approach it slightly differently by adding only rather than explicitly expanding −2:

Combining these identically as:
[tex]\[ -1.0 \cdot \ln (x) - \frac{1}{\ln (x)} \][/tex]

Therefore, \(-\frac{1}{\ln x} - \frac{1}{\ln x}\) simplifies to:
[tex]\[ -\frac{1}{\ln x} - \frac{1}{\ln x} = -1.0 \ln (x) - \frac{1}{\ln (x)} \][/tex]

### Summary

1. The expanded form of \(\ln \sqrt{\frac{x}{y}}\) is:
[tex]\[ \frac{1}{2} (\ln (x) - \ln (y)) \][/tex]

2. The simplified form of \(\frac{1}{- \ln x} - \frac{1}{\ln x}\) is:
[tex]\[ -1.0 \ln (x) - \frac{1}{\ln (x)} \][/tex]