If log 64 (base 10) is 1.8061, then what is the approximate value of log 16 (base 10)?

Introduction / Context:
This question requires using properties of logarithms to relate the log of one power of 2 to the log of another power of 2. We are given log 64 and asked to find log 16. Both numbers are powers of 2, which allows us to express them in terms of log 2 and connect the two values directly. This reinforces understanding of the power rule of logarithms.

Given Data / Assumptions:

• log 64 (base 10) = 1.8061.
• We need to find log 16 (base 10).
• 64 = 2⁶ and 16 = 2⁴.
• We use the rule log aⁿ = n log a.

Concept / Approach:
First write 64 and 16 as powers of 2: 64 = 2⁶ and 16 = 2⁴. From log 64 = log(2⁶) = 6 log 2, we can find log 2 by dividing the given value by 6. Then use log 16 = log(2⁴) = 4 log 2. Substituting the value of log 2 gives the required value of log 16, which can then be rounded to four decimal places and compared with the given options.

Step-by-Step Solution:
Step 1: Express 64 as a power of 2: 64 = 2⁶. Step 2: Using the power rule, log 64 = log(2⁶) = 6 log 2. Step 3: Given log 64 = 1.8061, we have 6 log 2 = 1.8061, so log 2 = 1.8061 / 6. Step 4: Compute log 2: 1.8061 / 6 ≈ 0.3010 (more precisely 0.3010167, close to the standard value). Step 5: Express 16 as 2⁴. Then log 16 = log(2⁴) = 4 log 2. Step 6: Substitute log 2 ≈ 0.3010: log 16 ≈ 4 × 0.3010 = 1.2040.

Verification / Alternative check:
We know from standard log tables that log 2 is about 0.3010, so log 16 = log(2⁴) = 4 × 0.3010 ≈ 1.2040. This matches our computed value. Also, 16 is between 10 and 100, so its log must be between 1 and 2. A value of 1.2040 is reasonable and consistent with 16 being closer to 10 than to 100.

Why Other Options Are Wrong:
Option 1.9048 is too large and would correspond to a number closer to 80 or 90, not 16. Option 0.9840 is less than 1 and would be for a number below 10. Option 1.4521 would be for a number around 28 to 30, also not 16. Option 1.0000 is exactly log 10 and does not match 16. Only 1.2040 fits both the power rule relationship and the approximate position of 16 on the log scale.

Common Pitfalls:
Some learners mistakenly divide by 4 instead of 6 when computing log 2, or they directly approximate log 16 without using the connection to log 64. Others confuse 2⁶ with 2⁴ and misapply the power rule. Being systematic, first finding log 2 from log 64 and then finding log 16, helps avoid these errors.

Final Answer:
The approximate value of log 16 is 1.2040 (base 10).