If log 2 = 0.3010 and log 3 = 0.4771 (to base 10), then what is the value of log base 5 of 512, that is log₅ 512?

Introduction / Context:
This problem tests the use of logarithm properties and change of base formula. We are given common logarithms (base 10) of 2 and 3 and asked to find log base 5 of 512. Since 512 is a power of 2 and 5 is related to 10 through 2, we can express everything in terms of base 10 logarithms, manipulate them using standard rules, and then compute the required value numerically.

Given Data / Assumptions:

• log 2 (base 10) = 0.3010.
• log 3 (base 10) = 0.4771 (though not directly needed for this question).
• We need to find log₅ 512.
• All unspecified logarithms are assumed to be base 10.
• We use the change of base formula: logₐ b = (log b) / (log a).

Concept / Approach:
First, note that 512 = 2⁹. So log 512 (base 10) equals 9 times log 2. Also, 5 can be written as 10 / 2, so log 5 can be expressed in terms of log 10 and log 2. The change of base formula tells us that log₅ 512 = (log 512) / (log 5), where these logs are base 10. After substituting the given value of log 2, we compute the numerical answer and compare it with the options given.

Step-by-Step Solution:
Step 1: Write 512 as a power of 2: 512 = 2⁹. Step 2: Compute log 512 using base 10: log 512 = log(2⁹) = 9 log 2 = 9 × 0.3010 = 2.709. Step 3: Express log 5 in terms of log 10 and log 2. Since 10 = 2 × 5, log 10 = log 2 + log 5. Thus, log 5 = log 10 − log 2 = 1 − 0.3010 = 0.6990. Step 4: Use change of base formula: log₅ 512 = (log 512) / (log 5) = 2.709 / 0.6990. Step 5: Divide 2.709 by 0.6990 to get approximately 3.8755, which rounds to 3.875.

Verification / Alternative check:
We can check whether 5³·⁸⁷⁵ is approximately equal to 512. Since 5³ = 125 and 5⁴ = 625, the true value should lie between 3 and 4, closer to 4. Our result 3.875 is indeed between 3 and 4 and closer to 4, which is reasonable. Also, using slightly more precise logs would give a value very close to 3.876, reinforcing that 3.875 is the correct option.

Why Other Options Are Wrong:
Option 2.875 is much too small and would correspond to a number far less than 512. Options 4.875 and 5.875 would lead to huge values, much greater than 512, because the exponential growth of 5 to such powers is very large. Only 3.875 matches the computed ratio of log 512 to log 5 using the given logarithm values.

Common Pitfalls:
Students sometimes forget to use the change of base formula and attempt to manipulate bases directly. Another common error is to miscalculate log 5; they might mistakenly add instead of subtracting, writing log 5 = 1 + 0.3010. Additionally, rounding too early in the process can lead to slight discrepancies; it is better to maintain at least three decimal places until the final division step.

Final Answer:
The value of log₅ 512 is 3.875.