The solutions of $\left(\frac{1}{2}\right)^{x}+\left(\frac{1}{6}\right)^{x}+\sqrt{2}\left(\frac{\sqrt{2}}{6}\right)^{x}=1$

Question

There is an algorithm to solve this equation? $$\left(\frac{1}{2}\right)^{x}+ \left(\frac{1}{6}\right)^{x}+\sqrt{2} \left(\frac{\sqrt{2}}{6}\right)^{x}=1$$

I can see that one solution is $x=1$ but I'm wondering if there're other solutions. Some hints?

Answer 1

Hello There is only one solution because left side of equation is decreasing function , right side is constant. Intersection of decreasing and constant functions have only one point. That point is (x,y)=(1,1)

Answer 2

The left side is the sum of three exponential functions, all of which are decreasing. So the left side always decreases. Its graph can cross the line $y=1$ at only one point.