Math Olympiad Problem | Nice Exponential Equation | Math Olympiad Training

[@SidneiMV]

xln5 + (1/x)(x - 1)ln8 = ln500
x²ln5 + (x - 1)ln8 - xln500 = 0
x²ln5 + xln(8/500) - ln8 = 0
x²ln5 - xln(500/8) - ln8 = 0
x²ln5 - xln(125/2) - 3ln2 = 0
x = [ln(125/2) ± √(ln²(125/2) + 12ln5ln2)]/(2ln5)
ln²(125/2) + 12ln5ln2 = ln²125 + ln²2 - 2ln125ln2 + 12ln5ln2 = ln²125 + ln²2 - 2ln125ln2 + 4ln125ln2 = (ln125 + ln2)²
x = [ln(125/2) ± (ln125 + ln2)]/(2ln5)
x = 2ln125/(2ln5) => *x = 3*
x = -2ln2/(2ln5) => *x = -ln2/ln5*

[@2012tulio]

X={3 ; -0.43}

[@MARTINWERDER]

@matholympiadonlinesolver
1/ 2 log base 5 = 5 log base 2 (= 2,321928) applied in the initial equation does not prove to be correct.
x = 3 and - 0,430676 (= negative 0,430676) are correct solutions and they give the result 500.