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To solve problem 18, which is to express the given quantity as a single logarithm, we will use the properties of logarithms.

The given expression is:
\[ \ln(a + b) + \ln(a - b) - 2 \ln c \]

We will use the following logarithm properties:
1. \(\ln(x) + \ln(y) = \ln(xy)\)
2. \(k \ln(x) = \ln(x^k)\)

First, combine the sum of the logarithms:
\[ \ln(a + b) + \ln(a - b) = \ln[(a + b)(a - b)] \]

Next, apply the subtraction of the logarithms:
\[ \ln[(a + b)(a - b)] - 2 \ln c \]

Rewrite \(2 \ln c\) using the property of logarithms:
\[ 2 \ln c = \ln(c^2) \]

So, we have:
\[ \ln[(a + b)(a - b)] - \ln(c^2) \]

Now, apply the property of logarithms for subtraction:
\[ \ln \left( \frac{(a + b)(a - b)}{c^2} \right) \]

Therefore, the expression \( \ln(a + b) + \ln(a - b) - 2 \ln c \) as a single logarithm is:
\[ \ln \left( \frac{(a + b)(a - b)}{c^2} \right) \]

Would you like more details or have any questions about this solution?

Here are some related questions to expand on this topic:
1. How do you combine multiple logarithms into a single logarithm using properties of logarithms?
2. What is the relationship between logarithms and exponents?
3. How do you solve equations involving logarithms?
4. Can you provide an example where logarithms are used to solve a real-world problem?
5. How do you differentiate a logarithmic function?
6. What is the change of base formula for logarithms?
7. How do logarithms relate to exponential growth and decay?
8. How do you solve inequalities involving logarithms?

**Tip:** When dealing with logarithms, always remember the basic properties and rules, such as the product, quotient, and power rules, as they are essential for simplifying and solving logarithmic expressions.

Learn how to combine multiple logarithms into a single logarithm using properties of logarithms. This detailed guide covers the logarithmic sum, subtraction, and power rules. Suitable for high school students studying logarithms and advanced algebra.

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