Calculate the roots of the equation 2^x+3^x=5^x?

We notice that the value x = 1 is the solution of the
equation.

 2^1 + 3^1 = 5^1

2 + 3 =
5

Now, we'll have to verify if x = 1 is the only solution for the
given equation or if there are more.

We'll divide the equation, both
sides by 5^x:

(2/5)^x + (3/5)^x =
1

We'll assign a function f(x) to the expresison (2/5)^x +
(3/5)^x.

The exponential functions (2/5)^x and (3/5)^x are
decreasing functions (the denominator is bigger than numerator), so f(x) is a decreasing
function, too.

If f(x) is a decreasing function, it could have only
one solution x = 1.

(2/5)^1 + (3/5)^1 =
1

(2+3)/5 = 1

5/5 =
1

1 = 1

So, x = 1 is the
only solution for the equation:

 2^x +
3^x = 5^x