Prove that 1 is the solution of the equation 5^x - 3^x = 2^x.Prove that 1 is the solution of the equation 5^x - 3^x = 2^x.
We only need to verify that 1 is the solution of the equation 5^x - 3^x = 2^x.
Any value raised to the power 1 gives the same value.
5^1 = 5, 3^1 = 3 and 2^1 = 2
5 - 3 = 2
Therefore x = 1 is the solution of 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.