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 10: If 3^{3 + 6 + 9 + ……… 3x} = (0.$\overline{037}$)^-66, what is the value of x?

1. 3
2. 6
3. 7
4. 11

Explanatory Answer

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

First of all, let us define the x^th term.
3^{3+6+9+……….3x} = (0.$\overline{037}$)^-66
Whenever you encounter a distinctive number such as one given in R.H.S of above equation, always try to find its significance in the context of question.

In this case L.H.S has 3^a, so (0.$\overline{037}$) must be some form of 3^a.
With little hit and trial, you may find $0.\overline{037}$ = $\frac{1}{27}\\$ = $\frac{1}{3^3}\\$ = 3^-3

3^{3(1 + 2 + 3 + ...X)} = 3 ^{-3 * -66}
=> 3³ * 3^x(x+1)/2 = 3^3*66
$\frac{x(x+1)}{2}\\$ = 66
x(x+1) = 132
Solving this equation for x > 0, we get x = 11.
You should directly be able to see that 132 = 11 * 12 => x= 11
And avoid wasting time solving the complete equation.

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

Hence, the answer is "11".

Choice D is the correct answer.