Let A = log base 32 of 1875 and B = log base 243 of 2187. Based on these definitions, which of the following relationships between A and B is correct?

Introduction:
This question tests your ability to compare two logarithmic expressions without necessarily computing them to many decimal places. By recognizing the structure of the numbers involved, you can simplify or estimate each logarithm and then decide whether A is less than, equal to, or greater than B.

Given Data / Assumptions:

• A = log base 32 (1875)
• B = log base 243 (2187)
• We must determine whether A < B, A = B, A > B, or if the relationship cannot be determined.

Concept / Approach:
The key idea is to exploit the fact that 32 and 243 are perfect powers and that 2187 is also a perfect power. Specifically, 32 = 2^5, 243 = 3^5, and 2187 = 3^7. For B, this allows an exact and simple calculation. For A, we can express 1875 in terms of prime factors and then obtain a reasonable estimate by comparing exponents, or by using change of base thinking qualitatively.

Step-by-Step Solution:
Step 1: Compute B exactly. 243 = 3^5 and 2187 = 3^7. Thus B = log base 3^5 (3^7) = 7 / 5 = 1.4. Step 2: Analyze A = log base 32 (1875). We know 32 = 2^5. Factor 1875: 1875 = 3 × 5^4. So A = log base 2^5 (3 × 5^4). This does not simplify to a simple rational like B, but we can compare orders of magnitude. Note that 2^5 = 32 and 32^2 = 1024, 32^3 = 32768. 1875 lies between 32^1 and 32^2, but much closer to 32^2 than to 32. As a result, log base 32 (1875) is between 1 and 2, and closer to 2 than to 1. A rough estimation shows A is a bit above 2, while B is exactly 1.4. Therefore, A is greater than B.

Verification / Alternative check:
Using approximate common logarithms, log10(1875) is about 3.27 and log10(32) is about 1.51. Then A ≈ 3.27 / 1.51 ≈ 2.16. Since B = 1.4 exactly, this confirms that A > B, with a comfortable margin so that rounding errors cannot change the ordering.

Why Other Options Are Wrong:
A < B would require A to be less than 1.4, which contradicts the fact that 1875 is far larger than 32 and close to 32^2. A = B is impossible because we have computed B exactly and A approximately as a clearly larger number. Saying the relationship cannot be determined is incorrect because we have enough information to compare them.

Common Pitfalls:
Some learners attempt to compare the arguments 1875 and 2187 directly, ignoring the different bases. Others may forget that larger bases shrink logarithm values and smaller bases increase them. Always rewrite numbers in terms of prime powers where possible and use the definition of logarithms logically rather than guessing.

Final Answer:
We conclude that A is greater than B, so A > B is correct.