prove that the equation $$2^x + 3^x + 4^x - 5^x =0$$
has just one root.
ATTEMPT: Write $2^x + 3^x + 4^x = 5^x$. By sketching the graphs it is confirmed that they will intersect at somewhere between $2$ & $3$. That's the only point in the first quadrant.
but can we prove that they will not intersect in the second quadrant?
Besides this can there be a more mathematical approach?