We have 5^x - 3^x = 2^x. Now to prove that x = 1 is a solution for the equation we substitute x = 1 and we should get the same value on both the sides

5^x - 3^x = 2^x

substituting x = 1 in the left hand side

=> 5^1 - 3^1 = 5 - 3 = 2

substituting x = 1 in the right hand side

=> 2^1 = 2

**Therefore as the left hand side is equal to the right hand side for x = 1, that is a solution.**

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.**

To prove that 1 is the solution of 5^x-3^x= 2^x.

To see wheher 1 is a solution to 5^x - 3^x = 2^x , we just put x= 1 and see whether it satisfies the equation 5^x-3^x = 2^x.

Left side 5^1 - 3^1 = 5-3 = 2.

Right side 2^1 = 2.

So the equation is satisfied as both sides are equal for x=1.

So x= 1 is a solution to 5^x-3^x = 2^x.