If x is a real number such that $l o g_{3} 5 = l o g_{5} (2 + x),$ then which of the following is true?

0 < x < 3

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23 < x < 30

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x > 30

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3 < x < 23

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Solution

The correct option is D
3 < x < 23

Best way to solve this question is by checking for different values.
$l o g_{3} 3 < l o g_{3} 5 < l o g_{3} 9$
$\Rightarrow 1 < l o g_{3} 5 < 2$
For x = 1, $l o g_{5} (2 + 1) = l o g_{5} 3,$ which is less than 1. Hence option (1) is eliminated.
For x = 25, $l o g_{5} (2 + 25) = l o g_{5} 27,$ which is more than 2. Hence option (2) and (3) are also eliminated.
Hence, correct answer is option (4).

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