Difficulty: Easy
Correct Answer: 1.2040
Explanation:
Introduction / Context:
This is a straightforward logarithm problem that uses the laws of indices and logarithms. You are given the common logarithm of 64 and asked to deduce the logarithm of 16. Since both numbers are powers of 2, the connection between them can be exploited to find log 16 without using tables or a calculator directly.
Given Data / Assumptions:
Concept / Approach:
We make use of the identity log(a^k) = k log a. Since 64 and 16 are powers of the same base, we can express both in terms of log 2 and relate log 16 to log 64. Express log 64 as 6 log 2, then find log 2 and use it to compute log 16 as 4 log 2.
Step-by-Step Solution:
Write 64 as 2⁶, so log 64 = log(2⁶) = 6 log 2.
We are given 6 log 2 = 1.8061.
Therefore, log 2 = 1.8061 / 6.
Divide: 1.8061 / 6 ≈ 0.3010167.
Next, write 16 as 2⁴. So log 16 = log(2⁴) = 4 log 2.
Substitute log 2: log 16 = 4 × 0.3010167 ≈ 1.2040668.
Rounded to four decimal places, log 16 ≈ 1.2040.
Verification / Alternative check:
We know that 10¹ = 10 and 10² = 100. Since 16 lies between 10 and 100, log 16 should lie between 1 and 2. A value of about 1.2 is therefore reasonable. This cross-check with the position of 16 on the number line confirms that 1.2040 is a sensible approximation.
Why Other Options Are Wrong:
1.9048 and 1.4521: These are too large for log 16, because they correspond to numbers much closer to 80 or 90 or beyond, not 16.
0.9840: This is less than 1, which would correspond to a number less than 10, whereas 16 is greater than 10.
Common Pitfalls:
A common mistake is to take a simple ratio of 16 and 64 and attempt to apply it directly to the logarithms without using the power rule. Another pitfall is rounding too early; dividing 1.8061 coarsely may introduce more error than necessary, although in this problem the answer options are far enough apart that small rounding differences do not affect the choice.
Final Answer:
The approximate value of log 16 is 1.2040.
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