A Beautiful Math Olympiad Exponential Equation | Nice Trick

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[@elmer6123]

Learn to recognize problems like this as Golden Ratio, Φ^2-Φ-1=0, with positive root, Φ=(√5+1)/2 or Inverse Golden Ratio, φ^2+φ-1=0, with positive root, φ=(√5-1)/2. Divide the given equation by 36^x and rearrange to [(7/6)^x]^2-(7/6)^x-1=0 and substitute Φ=(7/6)^x to obtain Φ^2-Φ-1=0, the Golden Ratio equation. Or divide the given equation by 49^x and rearrange to [(6/7)^x]^2+(6/7)^x-1=0 and substitute φ=(6/7)^x to obtain φ^2+φ-1=0, the Inverse Golden Ratio equation. Thus, x=lnΦ/ln(7/6) or x=lnφ/ln(6/7)

[@RealLifeTop10s]

nice question