Logs, What I thought was cool

I had previously learned that when typing log base Y (X) of different bases into my TI-83 I have type then in as Log base 10 (X)/Log bas 10 (Y).  But apparently it just compares the ratio of the original segments (X and Y).  So I could type in ln(X)/ln(Y)  or really any base over the same base of those two variables.  I found that quite logical and very cool too.