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Let's solve both parts of the problem by condensing the logarithmic expressions using properties of logarithms:

### Part (a): \( \log(6) + \log(x) + \log(y) \)
- The property of logarithms that allows us to combine the sum of logs into a single log is:  
  \[ \log(a) + \log(b) = \log(a \cdot b) \]
- Applying this to the expression:
  \[ \log(6) + \log(x) + \log(y) = \log(6 \cdot x \cdot y) \]
- Thus, the condensed expression is:
  \[ \log(6xy) \]

### Part (b): \( \ln(7) - \ln(x) - \ln(5) \)
- The property that allows us to combine the difference of logs is:  
  \[ \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \]
- Applying this step by step:
  \[ \ln(7) - \ln(x) = \ln\left(\frac{7}{x}\right) \]
  \[ \ln\left(\frac{7}{x}\right) - \ln(5) = \ln\left(\frac{7}{5x}\right) \]
- Thus, the condensed expression is:
  \[ \ln\left(\frac{7}{5x}\right) \]

Would you like further details or explanations?  

### Related questions:
1. How do you apply the properties of logarithms to more complex expressions?
2. Can the same properties be applied to natural logarithms (\(\ln\)) and common logarithms (\(\log\))?
3. What happens if there is a coefficient in front of the logarithms, such as \(2\log(x)\)?
4. How would the properties of logarithms change when dealing with subtraction of multiple logarithms?
5. What is the difference between \(\ln\) and \(\log\), and when should each be used?

### Tip: 
Remember that logarithmic properties can only be applied when the bases of the logarithms are the same (e.g., all \(\log\) or all \(\ln\)).

Condense each expression to a single logarithm using the properties of logarithms. a) log(6) + log(x) + log(y) b) ln(7) - ln(x) - ln(5)

Learn how to condense logarithmic expressions using the properties of logarithms. This step-by-step guide covers how to apply the product and quotient rules to combine multiple logarithmic terms into a single logarithm.

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