Why does lna - lnb = ln(a/b)?

It does not matter what base we use providing the same base is used for all logarithms, here we are using bease e.

Cerchiamo di definire A,B.C as follows=:

 A = ln a iff a = e^A ,

 B = ln b iff b = e^B

 C = ln (a/b) iff a/b = e^C

From the last definition we have:

 a/b = e^C => e^C = (e^A)/(e^B)

And using the law of indices:

 e^C = (e^A) (e^-B) = e^(A-B)

And as as the exponential is a 1:1 monotonic continuous function, we have:

 C = A-B

E così:

 ln (a/b) = ln a - ln b  QED