To express the given expression as a single logarithmic quantity, let's simplify the terms step-by-step. The expression is:

$\ln(9) + \frac{1}{2} \ln(x + 2) - 5 \ln(1 + \sqrt{x})$

Step 1: Apply the Power Rule for Logarithms

The power rule of logarithms states that $a \cdot \ln(b) = \ln(b^a)$. We can apply this rule to each term where there is a coefficient:

1. For $\ln(9)$, this term stays the same as it is already in a simple form.
2. For $\frac{1}{2} \ln(x + 2)$, we can rewrite it as: $\ln((x + 2)^{1/2}) = \ln(\sqrt{x + 2})$
3. For $-5 \ln(1 + \sqrt{x})$, we can rewrite it as: $\ln((1 + \sqrt{x})^{-5})$

Step 2: Combine the Terms Using Addition and Subtraction

Using the properties of logarithms, specifically:

• $\ln(a) + \ln(b) = \ln(a \cdot b)$
• $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$

We can combine the expression as follows:

$\ln(9) + \ln(\sqrt{x + 2}) - \ln((1 + \sqrt{x})^5)$

This simplifies further to:

$\ln \left( \frac{9 \cdot \sqrt{x + 2}}{(1 + \sqrt{x})^5} \right)$

Final Answer

The expression as a single logarithm is:

$\ln \left( \frac{9 \sqrt{x + 2}}{(1 + \sqrt{x})^5} \right)$

Would you like further details on any step, or do you have any questions?

Here are 5 questions to expand your understanding:

1. What are the properties of logarithms and how are they used in simplification?
2. How does the power rule of logarithms help in simplifying expressions?
3. Can this expression be simplified further if additional constraints are given on $x$?
4. What happens if we change the base of the logarithm?
5. How would this process change if the coefficients were different?

Tip: When combining logarithmic expressions, always consider the properties of logarithms such as product, quotient, and power rules.