Bonus Question for Some Logs These Are!

Algebra Level pending

What is the range of real value of A A such that log ⁡ y x + log ⁡ x y = A \log_{{y}}{{x}}+\log_{{x}}{{y}}=A has real solutions for x , y x,{y} ?


Notation: R \mathbb R denotes the set of real numbers .

Also try: Some Logs These Are!

R − ( − 2 , 2 ) \mathbb R - (-2,2) R − ( − 16 , 16 ) \mathbb R - (-16,16) R − ( 0 , 1 ) \mathbb R - (0,1) R − ( − 1 2 , 1 2 ) \mathbb R - \left (-\frac{1}{2},\frac{1}{2}\right)

Hua Zhi Vee by Hua Zhi Vee , 16, Malaysia

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1 solution

Hua Zhi Vee
Apr 9, 2017

Let

z = log ⁡ y x z = \log_{y}x

Use the quadratic formula on the equation

z 2 − A z + 1 = 0 z^{2} - Az +1 = 0

For real solution to this equation, range of real values A A can take is R − ( − 2 , 2 ) R - (-2,2)

(edited)

-- Yaşar Qamar

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