[wi] Bewijs: ln(ab) = ln(a) + ln(b)
 

Wie gaat deze oplossen:

http://www.steventenhave.nl/images/lnvraag.jpg

Log[ab] =
INT₁^ab (1/t) dt =
INT₁^a (1/t) dt + INT_a^ab (1/t) dt

Substitueer in de tweede integraal:
t = x/a
dt = dx/a
Grenzen:
t = a -> x = 1
t = ab -> x = b

Invullen:
INT_a^ab (1/t) dt =
INT₁^b (a/x) dx/a =
INT₁^b (1/x) dx

Oftewel:
INT₁^ab (1/t) dt = INT₁^a (1/t) dt + INT₁^b (1/x) dx
Log[ab] = Log[a] + Log[b]