When are logarithms transcendental?

When is the logarithm (of natural-number base) of a natural number rational, irrational, transcendental, etc? I'll leave the proof of our findings up to you! Follow me on Instagram for previews, behind-the-scenes, and more content!:   / epicmathtime   Support the channel on Patreon, I appreciate it a ton!:   / epicmathtime   Special thanks to my Channel Members and supporters on Patreon: Patreon Supporters: Dru Vitale, RYAN KUEMPER, AlkanKondo89, John Patterson, Johann Channel Members: Zach Ager Music: Fear Inoculum, Lateralus are originally by Tool, as performed by Sakis Strigas:    / tbird4pro   Please support Sakis for the amazing work he does! My current equipment for making videos (affiliate links): Camera: https://amzn.to/2HSJXDR Microphone(s): https://amzn.to/2SnpWY5 Audio Interface: https://amzn.to/3fcjMoc