CAT Quantitative Aptitude Questions | CAT Logarithms and Exponents

The question is from Log and Exponents. We can solve this easily using log identities. We need to find out the value of x from the given equation. With every extra hour you log in for this topic, it becomes exponentially simpler. CAT Logarithms and Exponents is a favorite in CAT Exam, and appears more often than expected in the CAT Quantitative Aptitude section in the CAT Exam

Question 7: log₃x + log_x3 = \\frac{17}{4}\\). Find the value of x.

1. 3⁴
2. 3^1/4
3. 3⁴ or 3^1/4
4. 3^1/3

Explanatory Answer

Method of solving this CAT Question from Logs and Exponents: When we change the base, Log questions become far tougher.

log₃x + log_x3 = \\frac{17}{4}\\)
Let y = log₃x
We know that log_x3 = \\frac{1}{log_3{x}}\\)
Hence log_x3 = \\frac{1}{y}\\)
Thus the equation can be written as y + \\frac{1}{y}\\) = \\frac{17}{4}\\)
4y² + 4 = 17y
4y² + 4 - 17y = 0
Solving the above equation we get y = 4 or \\frac{1}{4}\\)
If y = 4
log₃x = 4
Then x = 3⁴
If y = \\frac{1}{4}\\)
log₃x = \\frac{1}{4}\\)
Then x = 3^1/4

The question is "Find the value of x."

Hence, the answer is "3⁴ or 3^1/4".

Choice C is the correct answer.