Given that log 2 = 0.3010 and log 3 = 0.4771 (logarithms to base 10), find the value of log base 5 of 512, that is, log₅ 512.

Introduction / Context:
This question tests your understanding of logarithms, especially change of base and how to work with given common logarithm values. You are asked to compute log base 5 of 512 using only the given values log 2 and log 3 to base 10. This mimics typical exam conditions where you must manipulate logs algebraically rather than relying on a calculator.

Given Data / Assumptions:

• log 2 = 0.3010 (base 10).
• log 3 = 0.4771 (base 10).
• We need to find log₅ 512.
• Logarithms are common logs (base 10) when no base is specified.

Concept / Approach:
We use three main ideas. First, express 512 as a power of 2. Second, write log base 5 as a ratio of common logarithms using the change of base formula: log₅ N = (log N) / (log 5). Third, express log 5 in terms of log 2, because 5 = 10 / 2. This allows us to compute log 5 using 1 and log 2, which are known. Finally, substitute and simplify numerically to select the closest matching option.

Step-by-Step Solution:
Observe that 512 = 2⁹. Using the change of base formula, log₅ 512 = log 512 / log 5. Compute log 512 using powers of 2: log 512 = log(2⁹) = 9 log 2 = 9 × 0.3010 = 2.709. Next, find log 5. Since 10 = 2 × 5, we have log 10 = log 2 + log 5. log 10 = 1, so 1 = 0.3010 + log 5, giving log 5 = 1 - 0.3010 = 0.6990. Now compute log₅ 512 = 2.709 / 0.6990. The ratio 2.709 / 0.6990 is approximately 3.875 (close to 3.874, rounded).

Verification / Alternative check:
We know 5³ = 125 and 5⁴ = 625, so 512 lies between 5³ and 5⁴. Therefore log₅ 512 must lie between 3 and 4. Among the options, 3.875 is the only value between 3 and 4 and close to our calculated approximation, confirming our answer.

Why Other Options Are Wrong:
2.875: Less than 3, but 512 is larger than 125 = 5³, so the logarithm must be greater than 3.
4.875 and 5.875: These are larger than 4, which would correspond to numbers greater than 625 = 5⁴, so they cannot represent log₅ 512.

Common Pitfalls:
Students sometimes forget the change of base formula or mistakenly divide in the wrong order (log 5 / log 512). Another frequent error is computing log 5 incorrectly, for example using log 5 = log 2 + 1 instead of 1 - log 2. Rounding too early can also give slightly off answers, but the approximate range check between 3 and 4 usually corrects such issues.

Final Answer:
The required value is 3.875.