Poland | A nice Math Olympiad Exponent Simplification | Find all Roots of x ?

Poland | A nice Math Olympiad Exponent Simplification | Find all Roots of x ?#exponent#olympiad #simplification#exam
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[@rabotaakk-nw9nm]

After logarithm: xln5+x²ln19=ln5+ln19 => => x²+(ln5/ln19)x-(ln5/ln19+1)=0 => => x1=1; x2=-(ln5/ln19+1)≈-1.5466 😁

[@rainerzufall42]

The first solution is obviously x = 1. 5 * 19 = 95. Nice. After a little transform to (5 * 19^x)^x = 95, I can see a second easy solution: x = - log(95) / log(19) Why is that (BTW: log can be the logarithm to any base, let's say log = ln)? Well, 5 * 19^x = 5 * 95^-1 = 5/95 = 1/19, another power of 19. Now (19^-1)^x = 19^(+log(95)/log(19))= 95, the deserved result. Now before looking, what your video has to offer, the general approach: 5^x * 19^(x^2) = 19^(log(5)/log(19)*x)*19^(x^2) = 95 = 5 * 19 = 19^(log(5)/log(19)+1) Substitute y = log(5) / log(19), we have 19^(yx)*19^(x^2) = 19^(yx+xx) = 19^(y+1) Monotony of power function => yx + xx = y + 1. That's a start! x = 1 solves it. x = - (y + 1) = - log(19*5)/log(19) solves it: y(-y-1) + (-y-1)^2 = -y^2 - y + y^2 + 2y + 1 = -y + 2y + 1 = y + 1. qed

[@rainerzufall42]

For complex solutions, we need the complex log, which is periodic in 2 pi i. Let's omit this!

[@rainerzufall42]

Good job, you've found them. Just one remark: if you have 19^1 both in the products in the numerator and the denominator, you don't need to prove, that 19^(1-1) = 19^0 = 1. Just cancel them both!

[@rainerzufall42]

For the real solutions: yx+xx=y+1: x^2 + y x - y - 1 = 0 x = - y/2 +/- sqrt(y^2/4 + y + 1), but y^2/4 + y + 1 = ((y/2) + 1)^2, thus x = - y/2 +/- sqrt(((y/2) + 1)^2) = - y/2 +/- (y/2 + 1) = { 1, - y - 1 }, while - y - 1 = - log(5)/log(19) - log(19)/log(19) = - log(5*19)/log(19) = log_19(95) [I like your log_b(a) notation!]

[@CTJ2619]

obviously if x=1 then it works

[@roger7341]

By inspection x=1, but there is also a negative value. Try ln19*x^2+ln5*x-ln95=0, which has roots x=[-ln5±√(ln5^2+4*ln19*ln95)]/(2*ln19)=1 or -1.5466... Check 5^-1.5466*19^(1.5466^2)=95.0000....

[@rabotaakk-nw9nm]

Vieta's theorem for roots! 😅

[@umutkargili3617]

Please keep narrating instead of this noisy music

[@superacademy247]

Okay

[@powercloud5708]

Waste of paper