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Logarithmic Inequalities

log x 3-3/log...

Question

$\frac{l o g_{x} 3 - 3}{l o g_{x} 3 - 5} < 0$ .If a, b are integers which satisfy this inequation, find the maximum possible value of a - b.

A

214

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B

216

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C

200

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D

203

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Solution

The correct option is A
214

$\frac{l o g_{x} 3 - 3}{l o g_{x} 3 - 5} < 0$ .
Suppose $l o g_{x} 3 = y$
$\frac{y - 3}{y - 5} < 0 Y \in (3, 5) 3 < l o g_{x} 3 < 5$
27 < x < 243
Therefore max (a -b) will be when a = 242 and b = 28. Therefore, max (a -b) = 214.

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