WMC 2018 Senior Problem 13

Algebra Level 2

Solve over the set of real numbers: ∣ log ⁡ 5 ∣ x ∣ ∣ < 1 \left|\log_5{\left|x\right|}\right|<1 .

Notation: ∣ ⋅ ∣ |\cdot| denotes the absolute value function .

− 5 < x < 0 ∪ 0 < x < 5 -5<x<0 \cup 0<x<5 0 < x < 5 ∪ − 5 < x < 0 0<x<5 \cup -5<x<0 1 5 < x < 5 \frac{1}{5}<x<5 − 5 < x < 5 -5<x<5 − 5 < x < − 1 5 ∪ 1 5 < x < 5 -5<x<-\frac{1}{5} \cup \frac{1}{5}<x<5

Tiger Ang by Tiger Ang , 20, Malaysia

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1 solution

X X
Oct 14, 2018

− 1 < log ⁡ 5 ∣ x ∣ < 1 -1<\log_5 |x| <1

5 − 1 < ∣ x ∣ < 5 1 5^{-1}<|x|<5^1

So, − 5 < x < − 0.2 -5<x<-0.2 or 0.2 < x < 5 0.2<x<5

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