# CAT Practice : Exponents and Logarithms

Simple inequalities questions appear exponentially tougher when Logarithms are thrown in.

## Logarithm - Inequalities

Q.5: ${log_3{x-3} \over log_3{x-5}}$ < 0. If a, b are integers such that x = a, and x = b satisfy this inequation, find the maximum possible value of a – b.
1. 214
2. 216
3. 200
4. 203

Choice A. 214

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## Detailed Solution

${log_3{x - 3} \over log_3{x - 5}}$ < 0
log3x = y
${y - 3 \over y - 5}$ < 0
y ∈ (3, 5)
3 < log3x < 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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