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)} = log_{2}(9 - 2^{y}), Solve for y.

- 0
- 3
- 0 and 3
- none of these

none of these

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Before beginning to simplify the equation, don’t forget that anything inside a log cannot be negative

10^{log(3-y)} = log_{2}(9 - 2^{y}) (y > 0)…………………………………(1)

3 - y = log_{2}(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 –t^{2}

=> t^{2} - 9t + 8 = 0

=> t^{2} - 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."**

Choice D is the correct answer.

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