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, log_{36}x^{2} + log_{6}√y + 3log_{216}y^{1/2}z = 6. Find the value of x + y + z.

- 189
- 190
- 199
- 201

189

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Let us begin with simplifying the equation:-

=> log_{62}x^{2} + log_{6}y^{1/2} + 3log_{63}y^{1/2}z = 6

\\frac{2}{2}\\) * log_{6}x + \\frac{1}{2}\\) * log_{6}y + 3 * \\frac{1}{3}\\) * log_{6}y^{1/2}z = 6

log_{6}x + log_{6}y^{1/2}y^{1/2}z = 6

log_{6}x + log_{6}yz

log_{6}xyz = 6

xyz = 6^{6}

Given x,y,z is in G.P. Let x = a, y = ab, z = ab^{2}

=> xyz = a^{3}b^{3} = (ab)^{3}

(ab)^{3} = (6^{2})^{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."**

Choice A is the correct answer.

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