Find the logarithm of 144 to the base 23, that is evaluate log₂₃ 144. Choose the correct value from the given options.

Introduction / Context:
This question asks for log₂₃ 144, the logarithm of 144 to base 23. The options list a few integer values as well as a general option stating that none of these is correct. To answer correctly, we must reason about the magnitude of 23 raised to different powers in relation to 144, and we can also use the change of base formula for a more precise numerical estimate.

Given Data / Assumptions:

• The base is 23 and the argument is 144.
• We need to evaluate log₂₃ 144.
• Available options are 2, 4, 8, 16 and a general option None of these.
• All logarithms are real with positive bases and positive arguments, bases not equal to 1.

Concept / Approach:
First, use simple bounds. We know 23¹ = 23 and 23² = 529. Since 144 lies between 23 and 529, log₂₃ 144 must be between 1 and 2. Therefore it cannot be equal to 2, 4, 8, or 16, all of which are outside this interval. For additional confirmation, the change of base formula log₂₃ 144 = (log 144) / (log 23) in base 10 can be used to estimate its exact numerical value, which will clearly not be one of the integer options listed.

Step-by-Step Solution:
Step 1: Note that 23¹ = 23 and 23² = 529. Step 2: Since 144 is greater than 23 but less than 529, log₂₃ 144 must lie strictly between 1 and 2. Step 3: Observe that the options 2, 4, 8, and 16 are all greater than or equal to 2, so none of them can represent a logarithm value that is strictly between 1 and 2. Step 4: If we wish, we can apply the change of base formula: log₂₃ 144 = (log 144) / (log 23), where these logs are any common base such as 10. Step 5: Approximate values show that this ratio is a non integer real number between 1 and 2, confirming the earlier inequality based reasoning.

Verification / Alternative check:
We can further check by estimating the value numerically. Since 23² = 529 is much larger than 144, we expect log₂₃ 144 to be significantly closer to 1 than to 2. Using approximate decimal logs (for instance from a calculator) would confirm that log₂₃ 144 is approximately in the range 1.4 to 1.5, clearly not equal to any of the integer values 2, 4, 8, or 16.

Why Other Options Are Wrong:
Option 2 would imply 23² = 144, which is not true because 23² = 529. Options 4, 8, and 16 would imply enormous values like 23⁴, 23⁸, and 23¹⁶, which are far larger than 144. Therefore they cannot be the logarithm of 144 to base 23. The only reasonable conclusion is that the exact value is not among these integers.

Common Pitfalls:
Some learners may misunderstand and try to find an approximate integer close to the real value without checking the underlying power relationships. Others may think that since 144 is near 11², they must use 11 somehow, but the base here is 23, not 11. Always consider simple bounding by known powers of the base before attempting precise calculations.

Final Answer:
The correct choice is None of these, since log₂₃ 144 is a non integer between 1 and 2.