If log(x + 4) = log(4) + log(x) and log(y + 6) = log(6), compare x and y.
Aptitude
Logarithm
Difficulty: Easy
Choose an option
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Ax = y
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Bx < y
-
Cx > y
-
DCan't say
-
ENone of these
Answer
Correct Answer: x > y
Explanation
Introduction / Context:We solve two simple log equations independently and then compare the resulting values of x and y. The base is common (10 by default) and positivity of arguments must be ensured.
Given Data / Assumptions:
- log(x + 4) = log(4) + log(x) = log(4x).
- log(y + 6) = log(6).
- All log arguments are positive.
Concept / Approach:
- Use logarithm properties: log A + log B = log(AB).
- Since log is one-to-one on positive reals, equate arguments.
Step-by-Step Solution:
x + 4 = 4x ⇒ 3x = 4 ⇒ x = 4/3 ≈ 1.333…y + 6 = 6 ⇒ y = 0Compare: x ≈ 1.333… > y = 0 ⇒ x > y.Verification / Alternative check:Domain check: x + 4 > 0 and x > 0 (true for x = 4/3). For y, y + 6 > 0 (true for y = 0).
Why Other Options Are Wrong:
- x = y or x < y contradict the computed values.
- “Can’t say” is incorrect; both equations uniquely determine x and y.
Common Pitfalls:
- Using log(a) + log(b) = log(a + b), which is false.
Final Answer:x > y