How many digits does the number (9)^(9^9) have when written in base nine?

Question

How many digits does the number (9)^(9^9) have when written in base nine?

Answer 1

The number of digits in base 9 is based on logs

log[base 9](n) represents the number of digits n has. Therefore, we just take the log[base 9] of the number.

log[base 9](9^(9^9)) =
(9^9)log[base 9](9)

Since we know log[base 9](9) = 1, then our answer is simply

9^9

Which is

387420489

Answer 2

10^(10^10) has 10^10 digits in base 10

by analogy

9^(9^9) has 9^9 digits in base 9

or you can be a smart alec and write that the answer is 100,000,000 but you wrote the answer in base 9.