The minimum value of x in the domain

Algebra Level pending

Find the minimum possible value of x x such that log ⁡ ⌈ x − 1 ⌉ ⌊ x − 1 2 ⌋ \log_{\lceil x-1\rceil} \left \lfloor \dfrac{x-1}2\right \rfloor is a finite number.


The answer is 3.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

We must have ⌈ x − 1 ⌉ > 1 ⟹ x > 2 \lceil x-1\rceil >1\implies x>2 and ⌊ x − 1 2 ⌋ > 0 ⟹ x ≥ 3 \lfloor \dfrac{x-1}{2}\rfloor >0\implies x\geq 3 . Hence the minimum value of x x is 3 \boxed 3

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