Smeagol's Genius Game

Number Theory Level 2

Smeagol is a very brilliant creature! In his spare time, he figured out that \(\log_35 = 1.465\) without any calculators! With this brilliant knowledge, he decided to play a "game" with Bilbo Baggins, who is lost in Smeagol's cave. In this "game", Bilbo has to find the smallest \(b\) such that \(3^b > 5^{360}\). If Bilbo finds the correct \(b\), Smeagol must lead him out of the cave. Otherwise, Bilbo becomes Smeagol's lunch. What \(b\) should Bilbo choose to guarantee his safe passage out of the cave?

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