If log₈ x = 3/13 (logarithm to base 8), then what is the value of x? Choose the correct option from the following.

Introduction / Context:
This question involves solving a logarithmic equation for the unknown argument. We are given log₈ x = 3/13 and must find x. Instead of expecting a simple integer result, we must carefully apply the exponential form of the logarithm definition. The options include several integers and a general option None of these, so a key part of the solution is to check whether the derived value matches any of the integers given.

Given Data / Assumptions:

• The equation is log₈ x = 3/13.
• We require x to be positive, since logarithms of non positive numbers are not defined in the real system.
• The base 8 is positive and not equal to 1.
• We must choose the correct x among 25, 32, 37, 16 or None of these.

Concept / Approach:
From the definition of logarithm, log₈ x = 3/13 means 8^(3/13) = x. Thus we can express x directly as a fractional power of 8. By simplifying 8 as 2³ and carefully handling exponents, we can express x in terms of a power of 2 and then compare its size with the integer options. This will show whether x equals any of the proposed integers or not.

Step-by-Step Solution:
Step 1: Use the definition of logarithm. From log₈ x = 3/13, we have x = 8^(3/13). Step 2: Write 8 in terms of powers of 2: 8 = 2³. Step 3: Substitute into the expression for x: x = (2³)^(3/13). Step 4: Use the power of a power rule: (aᵐ)ⁿ = a^(m n). Here, x = 2^(3 × 3/13) = 2^(9/13). Step 5: The exponent 9/13 is a positive fraction less than 1, so x is greater than 1 but less than 2, because 2¹ is 2 and 2⁰ is 1. Step 6: Therefore x is a positive real number between 1 and 2 and clearly cannot be equal to 25, 32, 37, or 16, all of which are much larger than 2.

Verification / Alternative check:
We can approximate x numerically. Using a calculator or estimation, 8^(3/13) is roughly 1.616. Check against options: 25, 32, 37, and 16 are all much larger than 1.616. Since none of the given integers is close to this value, the correct answer cannot be any of these integers; the only remaining option is None of these.

Why Other Options Are Wrong:
If we suppose x = 25, then log₈ 25 would be roughly log 25 divided by log 8, which is not 3/13. Similarly, x = 32 would require 8^(3/13) to equal 32, but we know 8² = 64 and 8¹ = 8, so there is no exponent as small as 3/13 giving 32. The same reasoning eliminates 37 and 16. Each of these values gives a very different logarithm than 3/13 when checked directly.

Common Pitfalls:
Some learners guess that x must be a simple integer like 32 or 16 because powers of 2 are common, but they overlook that the exponent 3/13 is a small fraction. Others mistakenly solve 8^(13/3) instead of 8^(3/13). Carefully applying the rule logₐ b = c implies aᶜ = b is crucial for avoiding such errors.

Final Answer:
The correct choice is None of these, because x = 8^(3/13) = 2^(9/13), which is not equal to any listed integer option.