Prove that log(3) 7 is an irrational number

Proof below

Step-by-step explanation:

Irrational and Rational Numbers

We need to use the basic property of logarithms:

We are given:

And we will prove it's an irrational number, i.e., it's not a rational number.

Recall a rational number can always be expressed as a fraction a/b, where a and b are integer numbers. Thus, if the expression was a rational number, then:

Applying the property:

Raising to the power of b:

It's not possible to find two integers a and b that satisfy the condition above. That contradiction comes from the assumption that is rational, thus, it's proven is irrational.