A Challenging Problem Harvard Failed to Solve

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[@sunrazor2622]

3^x.5^x^2=15 e^(ln3.x).e^(ln5.x^2)=e^ln15 e^(ln3.x+ln5.x^2)=e^ln15 ln3.x+ln5.x^2=ln15 ln5.x^2+ln3.x-ln15=0 Use quadratic equation to solve: x = -0.5.ln3/ln5 ± 0.5.sqrt(ln3^2 + 4.ln5.ln15)/ln5 = 0.5.ln3/ln5 ±0.5.sqrt((ln3/ln5)^2 + 4.ln3/ln5 + 4)

[@sunrazor2622]

3^x.5^x^2=15 3^(x-1).5^(x^2-1)=1 ln3.(x-1)+ln5.(x^2-1)=0 ln3.x+ln5.x^2-ln3-ln5=0 ln3.x+ln5.x^2-ln15=0 Same result

[@sunrazor2622]

3^x.5^x^2=15 3^(x-1).5^(x^2-1)=1 3^(x-1).5^((x+1)(x-1))=1 ln3.(x-1)+ln5.(x+1)(x-1)=0 x=1 is one solution ln3+ln5.(x+1)=0 x= -ln3/ln5 -1 is the other solution