Difficulty: Easy
Correct Answer: 1/3
Explanation:
Introduction:
This question focuses on evaluating a simple logarithm where the base and the argument are powers of the same number. Recognizing this power relationship allows you to compute the logarithm exactly without using calculators or approximate values.
Given Data / Assumptions:
Concept / Approach:
When both the base and the argument of a logarithm are powers of the same number, we can write them as a^m and a^n, respectively. Then log base a^m (a^n) equals n / m. This comes directly from the definition of a logarithm and properties of exponents. Here, both 343 and 7 relate naturally to the base 7.
Step-by-Step Solution:
Step 1: Express 343 as a power of 7. 343 = 7 × 7 × 7 = 7^3. So the base 343 is 7^3, and the argument is 7^1. Step 2: Rewrite the logarithm using these powers. log base 343 (7) = log base 7^3 (7^1). Step 3: Use the identity log base a^m (a^n) = n / m. Here a = 7, m = 3, and n = 1. Therefore, log base 7^3 (7^1) = 1 / 3. So log base 343 (7) = 1/3.
Verification / Alternative check:
By definition, log base 343 (7) = x means 343^x = 7. If x = 1/3, then 343^(1/3) is the cube root of 343, which is 7. Since this matches the argument, x = 1/3 is indeed the correct logarithm value.
Why Other Options Are Wrong:
The value 3 would imply 343^3 = 7, which is false because 343^3 is much larger than 7. The values −3 and −1/3 would imply 343 raised to a negative power equals 7, which would yield a fraction less than 1, not 7. Therefore, none of these can satisfy the logarithm equation.
Common Pitfalls:
Learners sometimes reverse the roles of base and argument, accidentally evaluating log base 7 (343) instead of log base 343 (7). Others forget the exponent rule and try to approximate using decimal logs. Always rewrite both numbers as powers of the same base and apply the n / m rule carefully.
Final Answer:
The exact value of log base 343 (7) is 1/3.
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