proof verification - Prove $log_{4}6$ is irrational

Thanks for taking the time to verify my approach and as well as my answer.

Background:

• B.S. in Business from a 4-year university taking CS courses online


• I would like some help with a basic proof from MIT's 6.042J Mathematics for Computer Science course.

The question is:
Prove $\log_{4}6$ is irrational.

We prove the contradiction.

• Suppose $\log_{4}6$ is rational (i.e. a quotient of integers)
  $$\log_{4}6 = m/n$$


• So we must have m, n integers without common prime factors such that
  $$4^{m/n} = 6$$


• We will show that m and n are both even
  $$(4^{m/n})^{n} = 6^{n}$$


• So
  $$4^{m} = 6^{n}$$



• We then divide the two base numbers by their common factor, $2$, which gives us:

$$2^{m} = 3^{n}$$

• Since the product of two even numbers must be even AND the product of two odd numbers must be odd, $2^{m}$ and $3^{n}$ are not equivalent and therefore $m/n$ must not be rational.

Q.E.D. We conclude that $\log_{4}6$ is irrational.