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Some coinciding solutions of equations

Source: Romanian District Olympiad 2009, Grade X, Problem 3

October 8, 2018
equationsalgebralogarithms

Problem Statement

Let A A be the set of real solutions of the equation 3x=x+2, 3^x=x+2, and let be the set B B of real solutions of the equation log⁡3(x+2)+log⁡2(3x−x)=3x−1. \log_3 (x+2) +\log_2 \left( 3^x-x \right) =3^x-1 . Prove the validity of the following subpoints:
a) A⊂B. A\subset B. b) B⊄Q∧B⊄R∖Q. B\not\subset\mathbb{Q} \wedge B\not\subset \mathbb{R}\setminus\mathbb{Q} .
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