Proof that the Solution of 3^x = 8 is Irrational

In this post, we discuss another example of proof by contradiction. We are going to prove that the solution of the equation 3^x = 8 is irrational.

Theorem

The solution to the equation 3^x = 8 is irrational.

Proof

In proof by contradiction, we assume the opposite of the theorem and then find a contradiction somewhere in the proof.

Let us assume the opposite of the theorem. We assume that the solution to the equation 3^x = 8 is rational.

Now assuming that that the solution to the equation above is rational, then x is a rational number. This means that x =\frac{p}{q}, where p and q are integers and q not equal to 0 (definition of rational numbers). Therefore,

3^{\frac{p}{q}} = 8

Raising both sides to q, we have 3^p= 8^q.

Notice that this is a contradiction since 3^p is odd and 8^q is even and therefore the two numbers cannot be equal. In addition, 3^p = 2^{3q} contradicts the Unique Factorization Theorem or the Fundamental Theorem of Arithmetic.

Therefore, our assumption is false and the theorem that x is irrational is true.

 

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