اﻟﺪاﻟﺔ اﻷﺳﻴﺔ اﻟﺜﺎﻧﻴﺔ ﺳﻠﻚ ﺑﻜﺎﻟﻮرﻳﺎ ﻋﻠﻮم ﺗﺠﺮﻳﺒﻴﺔ -Іاﻟﺪاﻟﺔ اﻷﺳﻴﺔ اﻟﻨﻴﺒﺮﻳﺔ -1ﺗﻌﺎرﻳﻒ و ﺧﺎﺻﻴﺎت أوﻟﻴﺔ ﻧﻌﻠﻢ أن داﻟﺔ lnﺗﻘﺎﺑﻞ ﻣﻦ [∞ ]0; +ﻧﺤﻮ
و ﺑﺎﻟﺘﺎﻟﻲ ﺗﻘﺒﻞ داﻟﺔ ﻋﻜﺴﻴﺔ ﻣﻦ
ﻧﺤﻮ
[∞]0; +
أ -ﺗﻌﺮﻳﻒ اﻟﺪاﻟﺔ اﻟﻌﻜﺴﻴﺔ ﻟﺪاﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ اﻟﻨﻴﺒﻴﺮي ﺗﺴﻤﻰ اﻟﺪاﻟﺔ اﻷﺳﻴﺔ اﻟﻨﻴﺒﻴﺮﻳﺔ ﻧﺮﻣﺰ ﻟﻬﺎ )ﻣﺆﻗﺘﺎ( ﺑﺎﻟﺮﻣﺰ exp
exp ( x ) = y ⇔ ln y = x ب -ﺧﺎﺻﻴﺎت أوﻟﻴﺔ
exp ( 0 ) = 1
* * *
[∞∀y ∈ ]0; +
exp (1) = e
) exp ( x
∈ ∀x
ln ( exp ( x ) ) = x
∈ ∀x
0
* exp ( ln ( x ) ) = x
∈ ∀x
[∞∀x ∈ ]0; +
* اﻟﺪاﻟﺔ expﺗﺰاﻳﺪﻳﺔ ﻗﻄﻌﺎ ﻋﻠﻰ
*
exp ( a ) = exp ( b ) ⇔ a = b
2
∈ ) ∀ ( a; b
*
) exp ( a
2
∈ ) ∀ ( a; b
b
exp ( b ) ⇔ a
-2اﻟﺘﻤﺜﻴﻞ اﻟﻤﺒﻴﺎﻧﻲ ﻟﺪاﻟﺔ exp ﻓﻲ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ ﻣﻨﺤﻨﻰ اﻟﺪاﻟﺔ lnو ﻣﻨﺤﻨﻰ اﻟﺪاﻟﺔ expﻣﺘﻤﺎﺛﻼن ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﻤﻨﺼﻒ اﻷول
-3ﺧﺎﺻﻴﺔ أﺳﺎﺳﻴﺔ
) exp ( a + b ) = exp ( a ) × exp (b
اﻟﺒﺮهﺎن
2
∈ ) ∀ ( a; b
ln(exp ( a ) × exp ( b )) = ln exp ( a ) + ln exp ( b ) = a + b ln exp ( a + b ) = a + b ) ln ( exp ( a ) × exp ( b ) ) = ln exp ( a + b ) exp ( a + b ) = exp ( a ) × exp ( b
1