Russia Math Olympiad Problem | A nice Exponential Equation Simplification | Solve 10^x+100^x=1000^x

[@franciscook5819]

These questions seem simpler if you immediately use the relevant powers ... 10^x+100^x=1000^x (10^x)¹+(10^x)²=(10^x)³ multiply by 10^-x and rearrange all on one side (10^x)²-(10^x)¹-1=0 a quadratic in 10^x 10^x=(1±√(1+4))/2=(1±√5)/2 noting that √5>1 there is only one positive value ... x = log₁₀((1+√5)/2) (if you only have ln remember log₁₀(x)=ln(x)/ln(10) )

[@giannaleoci2328]

10^x+10^2x=10^3x 1+10^x=10^2x 10^x=y y^2-y-1=0 y=[1+-rq(1+4)]/2 y=[1+-rq5]/2 10^x=(1+-rq5)/2 1-rq5

[@solomonwilliams9708]

It can be further simplified by recalling that (1+√5)/2 = the golden ratio phi