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 y 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 12: 10^{log(3 - 10logy)} = log₂(9 - 2^y), Solve for y.

1. 0
2. 3
3. 0 and 3
4. none of these

Explanatory Answer

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

Before beginning to simplify the equation, don’t forget that anything inside a log cannot be negative
10^log(3-y) = log₂(9 - 2^y) (y > 0)…………………………………(1)
3 - y = log₂(9 - 2^y) (Therefore, 3 - y > 0 =) (y < 3)) ……………………… (2)
2^3-y = 9 - 2^y
2^y = t
\\frac{2^3}{t}\\) = 9 - t
=> 8 = 9t –t²
=> t² - 9t + 8 = 0
=> t² - t - 8t - 8 = 0
=> t(t - 1) - 8(t - 1) = 0
=> t = 1, 8
Therefore, 2^y = 1 and 2^y = 8
=> y = 0 and y = 3

However, from inequalities (1) and (2), y cannot take either of these value.

The question is "Solve for y."

Hence, the answer is "none of these".

Choice D is the correct answer.