Answer
See explanation
Work Step by Step
**Proof by Contradiction:**
Suppose, for the sake of contradiction, that \(\log_5 2\) is rational. Then there exist integers \(p\) and \(q\) with \(q > 0\) such that
\[
\log_5 2 = \frac{p}{q}.
\]
By the definition of logarithms, this equation is equivalent to
\[
5^{\frac{p}{q}} = 2.
\]
Raising both sides to the power \(q\) yields
\[
5^p = 2^q.
\]
Now, by the Fundamental Theorem of Arithmetic (unique prime factorization), the prime factors on the left must match those on the right. However, the left side, \(5^p\), is a power of 5 and contains no prime factors other than 5, while the right side, \(2^q\), is a power of 2 and contains no prime factors other than 2. This means that \(5^p\) cannot equal \(2^q\) unless both sides are 1, which is impossible because \(p\) and \(q\) are positive and \(2^q \ge 2\).
Thus, we have reached a contradiction.
Therefore, \(\log_5 2\) is irrational.
