Let’s analyze and solve the given expression:

$\log_{\sqrt{5} - 2} \left( \sqrt{5} + 2 \right) + \log_{\sqrt{3} - \sqrt{2}} \left( \sqrt{3} + \sqrt{2} \right).$

Step 1: Simplify the first term $\log_{\sqrt{5} - 2} (\sqrt{5} + 2)$

Using the logarithmic property:

$\log_a(b) = \frac{1}{\log_b(a)}.$ To simplify, we compute $(\sqrt{5} - 2)(\sqrt{5} + 2)$, which is a difference of squares: $(\sqrt{5} - 2)(\sqrt{5} + 2) = (\sqrt{5})^2 - (2)^2 = 5 - 4 = 1.$

This implies: $\sqrt{5} - 2 \text{ and } \sqrt{5} + 2 \text{ are reciprocal bases.}$ Thus: $\log_{\sqrt{5} - 2}(\sqrt{5} + 2) = 1.$

Step 2: Simplify the second term $\log_{\sqrt{3} - \sqrt{2}} (\sqrt{3} + \sqrt{2})$

Similarly, we compute $(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})$, which is also a difference of squares: $(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1.$

This implies: $\sqrt{3} - \sqrt{2} \text{ and } \sqrt{3} + \sqrt{2} \text{ are reciprocal bases.}$ Thus: $\log_{\sqrt{3} - \sqrt{2}} (\sqrt{3} + \sqrt{2}) = 1.$

Step 3: Combine the results

Now, we sum the results of the two logarithms: $\log_{\sqrt{5} - 2} (\sqrt{5} + 2) + \log_{\sqrt{3} - \sqrt{2}} (\sqrt{3} + \sqrt{2}) = 1 + 1 = 2.$

Final Answer:

$\boxed{2}$

Let me know if you would like further clarification!

Related Questions to Explore:

1. What are the properties of logarithms that can simplify expressions like this?
2. How does the difference of squares help in simplifying logarithmic bases?
3. How would the result change if the logarithmic bases were not reciprocal?
4. What are the implications of reciprocal bases for logarithmic calculations?
5. Can this problem be extended to complex logarithmic bases?

Tip:

When simplifying logarithms, always look for special relationships between the base and the argument, such as reciprocals or powers. This can save a lot of computation time!