# CAT Practice : Exponents and Logarithms

Basic Identities of Logarithms

## Basic identities of Logarithm

Q.11: x,y,z are 3 integers in a geometric sequence such that y- x is a perfect cube.
Given, log36x2 + log6√y + 3log216y1/2z = 6. Find the value of x+y+z.

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

Choice A. 189

## Detailed Solution

Let us begin with simplifying the equation:-

=> log62x2 + log6y1/2 + 3log63y1/2z = 6

$\frac{2}{2}$ * log6x + $\frac{1}{2}$ * log6y + 3 * $\frac{1}{3}$ * log6y1/2z = 6

log6x + log6y1/2y1/2z = 6

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

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