If , prove the…

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

If ln⁡ab−c=ln⁡bc−a=ln⁡ca−b\frac{\ln a}{b-c}=\frac{\ln b}{c-a}=\frac{\ln c}{a-b}, prove the following. (i) abc=1abc=1 (ii) aa⋅bb⋅cc=1a^a\cdot b^b\cdot c^c=1 (iii) ab2+bc+c2⋅bc2+ca+a2⋅ca2+ab+b2=1a^{b^2+bc+c^2}\cdot b^{c^2+ca+a^2}\cdot c^{a^2+ab+b^2}=1 (iv) a+b+c≥3a+b+c\ge3 (v) aa+bb+cc≥3a^a+b^b+c^c\ge3 (vi) ab2+bc+c2+bc2+ca+a2+ca2+ab+b2≥3a^{b^2+bc+c^2}+b^{c^2+ca+a^2}+c^{a^2+ab+b^2}\ge3

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