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