bonjour j'ai besoin de votre aide merci :)​

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)