Determine the solutions of $16^x + 81^x + 625^x = 60^x + 90^x + 150^x$.
By graphing this with Desmos I know the only solution is $0$ and I was able to spot that myself. Where my hardship lies is with proving that there are no more solutions. I thought of a few things that might help but turned out ineffective: constructing a function from that equation and differentiating to its find critical points but that leads to an even harder equation; dividing the whole equation by a certain term (e.g. $625$ or $60$) such that on one side I'd have a strictly increasing function and on the other a strictly decreasing function, leading to at most a single solution, approach which doesn't work due to $625$, the biggest base, being put together with those two other small numbers; and finally I noticed $150^x=(60+90)^x$ but I suspect $150$ is nearly a random number in this problem (not that my observation helps in any way).
That being said, how would one prove that there is only the unique solution $x=0$ for that equation? Any help and hints are much appreciated!