Réponse:
Bonjour
1.
log 1 = ln 1 / ln 10 = 0 / ln 10 = 0
log 10 = ln 10 / ln 10 = 1
log 100 = ln 100 / ln 10 = ln(10²)/ln(10) = 2ln10 / ln10 = 2
1b. log(10ᵖ) = ln(10ᵖ) / ln(10) = p×ln(10)/ln(10) = p
1c. log(x) = p => p = ln(x)/ln(10) => p×ln(10) = ln(x) => ln(10ᵖ) = ln(x) => (10ᵖ) = x
2a)
lim(ln(x))= -∞ et lim ln10 = ln 10
x→0 x→0
lim (log(x)) = lim [ln(x)/ln(10)] = -∞ par quotient des limites
x→0 x→0
lim(ln(x))= +∞ et lim ln10 = ln 10
x→+∞ x→+∞
lim (log(x)) = lim [ln(x)/ln(10)] = +∞ par quotient des limites
x→+∞ x→+∞
2b) log(x)' = 1/ln(10) × 1/x = 1/(xln10)
2c) Sur ]0;+∞[ , xln(10) > 0.
log'(x) > 0 donc la fonction log(x) est strictement croissante sur ]0;+∞[ .
2d
x |0 +∞
-----------------------------------
log'(x) || +
-----------------------------------
|| +∞
log(x) || ➚
|| -∞
2e. voir photo geogebra
3.
log(x) + log(y) = ln(x)/ln(10) + ln(y)/ln(10)
= [ln(x)+ln(y)]/ln(10)
= ln(xy)/ln(10)
= log(xy)
