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.