CAT Quantitative Aptitude Questions | CAT Logarithms and Exponents

The question is from Log and Exponents. This question is about logarithm inequalities. 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 13: 4^{6+12+18+24+…+6x} = (0.0625)^-84, what is the value of x?

1. 7
2. 6
3. 9
4. 12

Explanatory Answer

Method of solving this CAT Question from Logs and Exponents: Sum of N natural numbers comes as handy at times!

Take right side expression,
(0.0625)^-84 = \\frac{1}{16}\\)^-84
= (4^-2)^-84
= 4¹⁶⁸

Take left side expression
4^{6+12+18+24+…+6x} = 4^6(1+2+3+4+x)
= 4⁶ * 4^{(1+2+3+4+…+x)}
= 4⁶ * 4^x(x+1)/2 (using the formula for sum of natural numbers from 1 to x)
Equating left and right side expresssions, we get 4⁶ * 4^x(x+1)/2 = 4¹⁶⁸

Or 4⁶ * 4^x(x+1)/2 = 4^6*28
=> \\frac{x(x+1)}{2}\\) = 28
or x (x + 1) = 56
Solving for x we get, x = 7

The question is "what is the value of x?"

Hence, the answer is "7".

Choice A is the correct answer.