5. If logx (5/7) = −1/3, then the value of x is
(A) 343/125
(B) 125/343
(C) −25/49
(D) −49/25
Final Answer: x = 343/125
Given Equation
logx(5/7) = −1/3
Step-by-Step Solution
Step 1: Convert Logarithmic Form to Exponential Form
Using the definition of logarithms:
logx(a) = b ⇒ xb = a
Therefore:
x−1/3 = 5/7
Step 2: Take Reciprocal of Both Sides
Taking reciprocal to remove the negative exponent:
x1/3 = 7/5
Step 3: Cube Both Sides
Cubing both sides:
x = (7/5)3
x = 343/125
Final Answer
x = 343/125
Verification
Substitute x = 343/125: (343/125)−1/3=(125/343)1/3=5/7
Hence, the solution is verified.
Option Analysis
| Option | Value | Reason |
|---|---|---|
| (A) | 343/125 | Correct; equals (7/5)3 |
| (B) | 125/343 | Reciprocal gives a positive logarithm |
| (C) | −25/49 | Negative base not allowed in real logarithms |
| (D) | −49/25 | Negative value; logarithm undefined |
Introduction
The equation
logx(5/7) = −1/3
commonly appears in competitive exams such as GATE and JEE.
It tests understanding of logarithm rules, negative exponents,
and base restrictions.
Key Concept
If logb(a) = c, then bc = a.
A negative logarithmic value indicates that the argument is less than 1.
Exam Tips
- Always ensure base > 0 and base ≠ 1
- Argument of logarithm must be positive
- Negative logarithm values often lead to reciprocal forms
Conclusion
Solving logx(5/7) = −1/3 by converting to exponential form
gives the unique valid solution:
x = 343/125
This structured approach ensures accuracy and is ideal for
competitive mathematics examinations.
