For x in the natural numbers with x greater than 1, let p = log base x of (x + 1) and q = log base (x + 1) of (x + 2). Which of the following is correct?

Introduction / Context:
This problem compares two logarithmic expressions whose bases and arguments both depend on x. It is designed to test a deeper understanding of how logarithms behave when both the base and the argument change, and how to reason about inequalities involving logs. Rather than plugging in random values, the goal is to understand the monotonic behaviour and show that one expression is always larger than the other for natural x greater than 1.

Given Data / Assumptions:
- x is a natural number and x is greater than 1.
- p = log_x (x + 1).
- q = log_{x + 1} (x + 2).
- Bases x and x + 1 are greater than 1, so the logarithmic functions are increasing in their arguments.

Concept / Approach:
We convert both logarithms to a common base, for example base 10 or base e, using the change of base formula. Then we compare p and q numerically for general x. One useful idea is to express p and q in terms of ordinary logarithms of x, x + 1 and x + 2, and then analyse how the ratio of these logs behaves. Another more intuitive approach is to evaluate p and q for a few values of x (such as x = 2, 3, 4) and observe a consistent pattern, then reason why that pattern must hold for larger x as well.

Step-by-Step Solution:
Use the change of base formula: log_x (x + 1) = ln(x + 1) / ln x. Similarly, log_{x + 1} (x + 2) = ln(x + 2) / ln(x + 1). Therefore p = ln(x + 1) / ln x and q = ln(x + 2) / ln(x + 1). For x greater than 1, ln x, ln(x + 1) and ln(x + 2) are all positive. Consider x = 2 as a test value: p = ln 3 / ln 2, q = ln 4 / ln 3. Numerically, ln 3 / ln 2 is approximately 1.585, while ln 4 / ln 3 is approximately 1.262, so p > q at x = 2. For larger x, the ratio ln(x + 1) / ln x stays greater than 1 but slowly decreases, while ln(x + 2) / ln(x + 1) stays closer to 1 and is smaller than ln(x + 1) / ln x for all x greater than 1. This can be confirmed numerically or by more advanced analysis of the function f(x) = ln(x + 1) / ln x - ln(x + 2) / ln(x + 1), which remains positive for x greater than 1. Hence p is greater than q for all natural x greater than 1.

Verification / Alternative check:
Evaluating for several natural values gives a clear pattern. For x = 3, p = ln 4 / ln 3 which is about 1.262, and q = ln 5 / ln 4 which is about 1.161, so p > q again. For x = 4, p is about 1.161, while q is about 1.113. This consistent behaviour suggests that p always exceeds q. The trend arises because as the base grows, the relative increase from x to x + 1 is larger than the relative increase from x + 1 to x + 2 in logarithmic terms.

Why Other Options Are Wrong:
p < q: This contradicts both numerical checks and the analytical behaviour of the expressions.
p = q: There is no natural x greater than 1 for which ln(x + 1) / ln x equals ln(x + 2) / ln(x + 1).
cannot be determined: The relationship can be determined and is consistently p > q for all natural x greater than 1.

Common Pitfalls:
Some students may incorrectly think that since x, x + 1 and x + 2 are close, p and q should be equal or impossible to compare. Others may forget that when the base of a logarithm is greater than 1, the function is increasing in its argument, which affects the inequality direction. Another frequent mistake is to attempt to compare p and q by cross multiplying the logs without taking care of the domains and positivity, which can lead to sign errors. Using the change of base formula in a systematic way and checking a few concrete numerical examples helps build correct intuition.

Final Answer:
For all natural numbers x greater than 1, we have p > q.