A Nice Olympiad Math Algebra Problem | How to solve for x?

[@edsznyter1437]

This seems set up for the Lambert function. 27^(x-3) = 3-x 1 = (3-x)e^((3-x)ln27 ln27 = (3-x)ln27*e^((3-x)ln27) ln3 = W(ln3 e^ln3) = W(3ln3) = W(ln 27) = W((3-x)ln27*e^((3-x)ln27) = (3-x)ln27 = 3(3-x)ln3 1 = 3(3-x) x = 8/3

[@SALogics]

Very nice! ❤

[@zawatsky]

3^3(x-3)=3-x. a:=x-3. 3^3a=-a. 3^3=(-a)^(1/a)=(-1/a)^(-1/a). -1/a=3. a=-1/3=x-3. x=3-1/3=2²/₃. Подставим все переменные в исходное: 27^a+x=27^(-1/3)+2²/₃=3. ³√(1/27)=3-2²/₃=1/[³√(27)]=1-²/₃=¹/₃.

[@فیروزاهنگری]

27^(x-3)=3-x [27^(x-3)]^1/(x-3)=(3-x)^1/(x-3) 27=(3-x)^1/(x-3) (1/3)^1/(-1/3)=(3-x)^1/(x-3) 1/3=3-x , x=8/3

[@SALogics]

Very nice! ❤

[@prollysine]

3*ln3=3*ln3(3-x)*e^(3*ln3(3-x)) , W(ln3*e^(ln3))=W(3*ln3(3-x)*e^(3*ln3(3-x))) , ln3=3*ln3(3-x) , 1=3*(3-x) , x=9/3 - 1/3 , x=8/3 , W(3*ln3)=3*ln3(3-x) , 3-(W(3*ln3)/3*ln3)=x , x=8/3 , same ,

[@SALogics]

Very nice! ❤

[@prollysine]

@@SALogics Thanks!