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 + y + z 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 11: x, y, z are 3 integers in a geometric sequence such that y - x is a perfect cube. Given, log₃₆x² + log₆√y + 3log₂₁₆y^1/2z = 6. Find the value of x + y + z.

1. 189
2. 190
3. 199
4. 201

Explanatory Answer

Method of solving this CAT Question from Logs and Exponents: Application of Basic Identities of Logarithms!

Let us begin with simplifying the equation:-
=> log_6²x² + log₆y^1/2 + 3log_6³y^1/2z = 6
\\frac{2}{2}\\) * log₆x + \\frac{1}{2}\\) * log₆y + 3 * \\frac{1}{3}\\) * log₆y^1/2z = 6
log₆x + log₆y^1/2y^1/2z = 6
log₆x + log₆yz
log₆xyz = 6
xyz = 6⁶

Given x,y,z is in G.P. Let x = a, y = ab, z = ab²
=> xyz = a³b³ = (ab)³
(ab)³ = (6²)³

Possible values of (a,b) satisfying the equation :-
(1, 36), (2, 18), (3, 12), (4, 9), (9, 4), (12, 3), (18, 2), (36, 1)
Given y-x is a perfect cube
=> ab-a is perfect cube
=> a(b-1) is perfect cube

Only possible when (a, b) = (9, 4)
∴ x = 9 , y = 36 , z = 144
∴ x + y + z = 9 + 36 + 144 = 189

The question is "Find the value of x + y + z."

Hence, the answer is "189".

Choice A is the correct answer.