If log₇ 2 = m (logarithm to base 7), then what is the value of log₄ 28 in terms of m?

Introduction / Context:
This question tests the ability to express one logarithm in terms of another using properties of logarithms and change of base. We are given log₇ 2 in terms of a parameter m and asked to find log₄ 28 in terms of m. The key is to rewrite 28 and 4 using prime factorisation and then apply log rules carefully to relate them back to m.

Given Data / Assumptions:

• log₇ 2 = m.
• We need to express log₄ 28 in terms of m.
• All logs are real and defined for positive arguments and bases not equal to 1.
• Note that 28 = 4 × 7 and 4 = 2².

Concept / Approach:
Since 28 = 4 × 7, we can write log₄ 28 = log₄ (4 × 7) = log₄ 4 + log₄ 7 = 1 + log₄ 7, because log₄ 4 is 1. Therefore we only need log₄ 7. Using the change of base formula log₄ 7 = (log 7) / (log 4) and expressing both numerator and denominator in terms of log 2 and log 7, we can then connect back to the given relation log₇ 2 = m and simplify to get an expression involving m only.

Step-by-Step Solution:
Step 1: Start from log₄ 28. Since 28 = 4 × 7, we have log₄ 28 = log₄ (4 × 7) = log₄ 4 + log₄ 7. Step 2: log₄ 4 equals 1, so log₄ 28 = 1 + log₄ 7. Step 3: Use change of base for log₄ 7: log₄ 7 = (log 7) / (log 4), where logs are base 10 or any common base. Step 4: Express log 4 in terms of log 2: log 4 = log(2²) = 2 log 2. Step 5: From log₇ 2 = m, using change of base, log₇ 2 = (log 2) / (log 7) = m. Therefore, log 2 = m log 7. Step 6: Substitute log 2 = m log 7 into log 4: log 4 = 2 log 2 = 2m log 7. Step 7: Now log₄ 7 = (log 7) / (log 4) = (log 7) / (2m log 7) = 1 / (2m). Step 8: Finally, log₄ 28 = 1 + log₄ 7 = 1 + 1 / (2m) = (2m + 1) / (2m).

Verification / Alternative check:
As a numerical check, suppose log₇ 2 equals a concrete value by taking 2 and 7 logs in base 10. Then compute m = log 2 / log 7, and separately compute log₄ 28 numerically. Simplifying (2m + 1) / (2m) and comparing with the numeric log₄ 28 confirms that the derived expression is correct, since both evaluations match up to rounding errors.

Why Other Options Are Wrong:
The other algebraic expressions such as 1/(1 + 2m), (1 + 2m)/2, 2m/(2m + 1), and 4m/(2m + 1) do not follow from the change of base derivation. Substituting specific approximate values for m put them in clear disagreement with the actual value of log₄ 28, while (2m + 1) / (2m) agrees exactly.

Common Pitfalls:
One common mistake is to attempt to treat log₇ 2 as log 7 divided by log 2, which is the reciprocal of the correct change of base relation. Another error is to misinterpret 28 as 2 × 7² rather than 4 × 7 and then take a longer and more error prone path. Careful use of prime factorisation and attention to base conversions greatly simplifies these problems.

Final Answer:
The value of log₄ 28 in terms of m is (2m + 1) / (2m).