- #1
jgens
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Homework Statement
Prove that [itex]\log_{10}(2)[/itex] is irrational.
Homework Equations
N/A
The Attempt at a Solution
Suppose not, then [itex]\log_{10}(2) = p/q[/itex] where p and q are integers. This implies that [itex]2 = 10^{p/q}[/itex] or similarly, [itex]2^q = 10^p[/itex]. However, this is a contradiction since each number's prime factorization is unique - [itex]2^q[/itex] contains only 2's as prime factors while [itex]10^p[/itex] contains both 2's and 5's. Therefore, our assumption that [itex]\log_{10}(2)[/itex] was rational must have been incorrect. This completes the proof.
I'm really bad at these irrationality proofs so I was wondering if someone could comment on the validity of my method. Thanks!