# 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.

justaguide | College Teacher | (Level 2) Distinguished Educator

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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.

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.