Prove that 1 is the solution of the equation 5^x-3^x=2^x

We notice that substituting x by the value 1, we'll verify the
equation, so x = 1 is the solution of the equation.

2^1 = 5^1 -
3^1

2 = 5 -  3

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 the greatest
exponential, namely 5^x:

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

We'll put f(x) = 1 -
(3/5)^x

We'll calculate f(1):

2/5 = 1 -
3/5 

2/5 = (5-3)/5

2/5 =
2/5

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.