# CAT Practice : Exponents and Logarithms

Basic Identities of Logarithms

## Basic identities of Logarithm

Q.10: If 33+6+9+………3x = (0.$\overline{037}$)-66, what is the value of x?
1. 3
2. 6
3. 7
4. 11

Choice D. 11

## Detailed Solution

First of all, let us define the xth term.
33+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 3power, so (0.$\overline{037}$) must be some form of 3power.
With little hit and trial, you may find $0.\overline{037} = \frac{1}{27} = \frac{1}{3^3} = 3^-3$
33(1 + 2 + 3 + ...X) = 3 -3 * -66
=> $3^3 * 3^\frac{x(x+1)}{2} =$33*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 x 12 =) x= 11
And avoid wasting time solving the complete equation.

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