The golden ratio rounded up is 2. QED.
“100 in decimal has three digits, so its radix economy is 10×3 = 30; its binary representation has seven digits (1100100 2) so it has radix economy 2×7 = 14 in base 2; in base 3 its representation has five digits (10201 3) with a radix economy of 3×5 = 15; in base 36 (2S 36) its radix economy is 36×2 = 72.” https://en.wikipedia.org/wiki/Radix_economy
The idea being that there’s a tradeoff between the cost of each digit vs the number of digits you need. However we use base 2 because the cost of base 3 is more than 50% higher than using base 2.