Explain why 321 base 5 and 321 base 6 are not the same WITHOUT translating into Base 10?

Question

You guys translated it into Base 10.

Answer 1

N = 321_6 - 321_5
Each digit in 321_6 has a higher value than the corresponding digit in 321_5, thus N > 0.

Answer 2

Each 321 represents 3 in the base-squared place, 2 in the base place, and 1 in the units place. Since the bases are different, the numbers are different.

Answer 3

let Log of 321 to base 6 be =y then 6^y=321. Now take log of both sides to base 5.
We get y* log 6 to base 5= log 321 to base 5
Now then y= log 321 to base 5/ log 6 to base 5, but y is equal to log 321 to base 6(first statement)

Hence log321 to base 6= log 321 to base 5/ log 6 to base 5. hence 321 base6 is not equal to 321 base 5