What are the roots of the equation 2^{x+2} , 27^{x/x-1} = 9 ?

Correct Answer is : - 2,1 -(log⁡3log⁡2)≤ft( 3/ 2 )(log2log3)

Correct Answer is : - 2,1 -(log⁡3log⁡2)≤ft( 3/ 2 )(log2log3)

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CAT Exam Model Paper 2 with solutions for free online practice

Question : 91 of 100

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2^{x+2} , 27^{\frac{x}{x-1}} = 9 ?

Solution:

2^{x+2} , 27^{\frac{x}{x-1}} = 9

On taking log both sides, we get

(x+2) \log 2 + \frac{x}{x-1} \log 27 = \log 9

\Rightarrow (x+2) \log 2 + \frac{3x}{x-1} \log 3 = 2 \log 3

\Rightarrow (x+2) \log 2 + \left( \frac{3x}{x-1} - 2 \right) \log 3 = 0

\Rightarrow (x+2) \log 2 + \left( \frac{x+2}{x-1} \right) \log 3 = 0

\Rightarrow (x+2) \left[ \log 2 + \frac{1}{x-1} \log 3 \right] = 0

\Rightarrow x + 2 = 0

\log 2 + \frac{1}{x-1} \log 3 = 0

\Rightarrow x = -2

(x-1) = - \frac{\log 3}{\log 2}

\therefore x = -2

x = 1 - \left( \frac{\log 3}{\log 2} \right)

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