[wi] Bewijs: ln(ab) = ln(a) + ln(b)

**03-01-2005, 15:20**

Wie gaat deze oplossen:

GinnyPig · **03-01-2005, 16:31**

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]

03-01-2005, 16:31
GinnyPig	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] __________________ O_o

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