The value of the logarithm log base 343 of 7, that is log_343 7, is equal to which of the following?

Introduction / Context:
This question checks understanding of logarithms with unusual bases and arguments. Instead of simple base 10 or base e, the base is 343 and the argument is 7. Problems like this appear in aptitude tests to see whether the candidate can recognize powers and rewrite a logarithm in terms of exponents without relying on a calculator.

Given Data / Assumptions:
We are asked to find log base 343 of 7.The expression is written as log_343 7.The usual change of base and exponent properties of logarithms are available.We work with real positive numbers only.

Concept / Approach:
The key idea is to express both the base and the argument as powers of the same prime number. The number 343 can be written as 7^3. The number 7 is 7^1. When both base and argument share the same prime factor, the logarithm log_(a^k) (a^m) simplifies neatly to m / k. This comes from the definition of logarithm: log_b x is the exponent to which b must be raised to produce x.

Step-by-Step Solution:
Step 1: Rewrite the base 343 as a power of 7: 343 = 7^3.Step 2: Rewrite the argument 7 as 7^1.Step 3: Now consider log_343 7 as log_(7^3) (7^1).Step 4: By the definition of logarithm, we seek the exponent y such that (7^3)^y = 7^1.Step 5: Simplify the left side: (7^3)^y = 7^(3y).Step 6: Equate exponents since the bases are equal: 3y = 1.Step 7: Solve for y: y = 1 / 3.Step 8: Therefore log_343 7 = 1 / 3.

Verification / Alternative check:
We can confirm the result by using the change of base formula. Write log_343 7 = log 7 / log 343, where log denotes any consistent base, for example base 10. Then express 343 as 7^3. Hence log 343 = log (7^3) = 3 log 7. So log_343 7 = log 7 / (3 log 7) = 1 / 3. This matches the earlier exponent based reasoning and confirms that the correct value is one third.

Why Other Options Are Wrong:
The option 3 would correspond to log_7 343, not log_343 7. The option −3 and the option −1 / 3 are negative, but both base and argument are greater than 1, so the logarithm must be positive. Therefore, all negative values are impossible. Only 1 / 3 fits the algebraic derivation and the general behavior of logarithms.

Common Pitfalls:
A common error is to invert the roles of base and argument and mistakenly compute log_7 343. Another mistake is to overlook that 343 is a power of 7 and attempt to use approximate decimal values, which complicates the problem unnecessarily. Recognizing power relationships and using exponent rules keeps the solution short and accurate.

Final Answer:
The value of log base 343 of 7 is 1/3, so the correct option is 1/3.