Find the logarithm of 144 to base 23, that is, evaluate log₂₃ 144 and choose the closest correct option.

Introduction / Context:
This problem is about estimating the value of a logarithm with an unusual base. Instead of asking for a detailed decimal approximation, the question simply wants to know which listed option could be equal to log₂₃ 144. You can solve this by comparing powers of 23 with 144 without using a calculator.

Given Data / Assumptions:

• We need log₂₃ 144.
• Base is 23, argument is 144.
• Available options are 2, 4, 8, and None of these.
• All numbers are positive, and the logarithm is real.

Concept / Approach:
The logarithm log₂₃ 144 answers the question, "To what power must 23 be raised to obtain 144?" Therefore, we should compare 23¹, 23², and nearby powers to see the range in which 144 falls. If 144 were exactly a simple power of 23, its logarithm would be an integer, but otherwise it will be some non-integer value between two integers. This reasoning alone is enough to choose the correct option from the list.

Step-by-Step Solution:
Start with 23¹ = 23. Next, compute 23² = 23 × 23 = 529. Now compare 144 with these powers. We see that 23 < 144 < 529. Therefore, 23¹ < 144 < 23². This implies 1 < log₂₃ 144 < 2, because logarithms are increasing functions for bases greater than 1. Hence log₂₃ 144 is not equal to 2, 4, or 8, all of which are integers outside the interval (1, 2).

Verification / Alternative check:
If log₂₃ 144 were exactly 2, then 23² would equal 144, but 23² is 529. If it were 4, then 23⁴ would need to equal 144, but 23⁴ is huge, far above 144. Thus, any integer value from the options is clearly impossible.

Why Other Options Are Wrong:
2: Would require 144 = 23² = 529, which is false.
4: Would require 144 = 23⁴, which is vastly larger than 144.
8: Would require 144 = 23⁸, an even more enormous number, so this is clearly incorrect.

Common Pitfalls:
Sometimes students try to overcomplicate such questions by attempting long logarithm calculations without real need. Here, conceptual understanding of logarithm behaviour and simple power comparisons are enough. Another mistake is to assume that because the number 144 is a perfect square (12²), it must somehow match a simple integer logarithm base 23, which is not true.

Final Answer:
Therefore, none of the integer values listed match log₂₃ 144. The correct choice is None of these.