Finding all real solutions to the equation $3^x+4^x=5^x$
Hint: Divide through by $5^x$ to produce the equivalent equation $$\left(\frac{3}{5}\right)^x + \left(\frac{4}{5}\right)^x = 1.$$ The l.h.s. is a strictly decreasing function of $x$.
Hint: Divide through by $5^x$ to produce the equivalent equation $$\left(\frac{3}{5}\right)^x + \left(\frac{4}{5}\right)^x = 1.$$ The l.h.s. is a strictly decreasing function of $x$.