Why does lna−lnb=ln(ab)?
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=lna⇔a=eA,
B=lnb⇔b=eB
C=ln(ab)⇔ab=eC
From the last definition we have:
ab=eC⇒eC=eAeB
And using the law of indices:
eC=(eA)(e−B)=eA−B
And as as the exponential is a 1:1 monotonic continuous function, we have:
C=A−B
E così:
ln(ab)=lna−lnb QED