‫داﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ‬

‫اﻟﺜﺎﻧﻴﺔ ﺳﻠﻚ ﺑﻜﺎﻟﻮرﻳﺎ ﻋﻠﻮم رﻳﺎﺿﻴﺔ‬

‫‪ -І‬داﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ اﻟﻨﻴﺒﻴﺮي‬

‫‪ -1‬ﺗﺬآﻴﺮ ‪ -‬ﻧﻌﻠﻢ أن آﻞ داﻟﺔ ﻣﺘﺼﻠﺔ ﻋﻠﻰ ﻣﺠﺎل ‪ I‬ﺗﻘﺒﻞ دوال أﺻﻠﻴﺔ ﻋﻠﻰ ‪I‬‬
‫‪x r +1‬‬
‫اﻟﺪاﻟﺔ ‪ x →x‬ﺗﻘﺒﻞ دوال أﺻﻠﻴﺔ ﻋﻠﻰ [∞‪ ]0; +‬هﻲ ‪+ k‬‬
‫‪r‬‬
‫→ ‪x‬‬ ‫‪ -‬ﻧﻌﻠﻢ أن ﻟﻜﻞ ‪ r‬ﻣﻦ }‪− {−1‬‬
‫‪r +1‬‬
‫ﺣﻴﺚ ‪ k‬ﻋﺪد ﺣﻘﻴﻘﻲ ﺛﺎﺑﺖ‬
‫‪1‬‬
‫اﻟﻤﺘﺼﻠﺔ ﻋﻠﻰ [∞‪ ]0; +‬وﻣﻨﻪ ﺗﻘﺒﻞ دوال أﺻﻠﻴﺔ‬ ‫→‪x‬‬
‫*‪ -‬ﻓﻲ اﻟﺤﺎﻟﺔ اﻟﺘﻲ ﺗﻜﻮن ‪ r=-1‬ﻧﺤﺼﻞ ﻋﻠﻰ اﻟﺪاﻟﺔ‬
‫‪x‬‬
‫‪1‬‬
‫وﺑﺎﻟﺘﺎﻟﻲ اﻟﺪاﻟﺔ → ‪ x‬ﺗﻘﺒﻞ داﻟﺔ أﺻﻠﻴﺔ وﺣﻴﺪة ﺗﻨﻌﺪم ﻓﻲ ‪.1‬‬
‫‪x‬‬
‫‪ -2‬ﺗﻌﺮﻳﻒ‬
‫‪1‬‬
‫اﻟﺪاﻟﺔ اﻷﺻﻠﻴﺔ ﻟﺪاﻟﺔ → ‪ x‬ﻋﻠﻰ [ ∞ ‪ ]0; +‬اﻟﺘﻲ ﺗﻨﻌﺪم ﻓﻲ اﻟﻨﻘﻄﺔ ‪ 1‬ﺗﺴﻤﻰ داﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ اﻟﻨﻴﺒﻴﺮي‬
‫‪x‬‬
‫و ﻳﺮﻣﺰ ﻟﻬﺎ ﺑﺎﻟﺮﻣﺰ ‪ ln‬أو ‪Log‬‬
‫‪ x 0‬‬
‫‪‬‬
‫‪‬‬ ‫‪1‬‬
‫) ‪f ' ( x ) = ⇔ f ( x ) = ln(x‬‬
‫‪‬‬ ‫‪x‬‬
‫‪ f (1) = 0‬‬

‫‪ -3‬ﺧﺎﺻﻴﺎت‬
‫أ‪ -‬ﺧﺎﺻﻴﺎت‬
‫‪ln(1)=0‬‬ ‫[∞‪]0; +‬‬ ‫*‪ -‬ﻣﺠﻤﻮﻋﺔ ﺗﻌﺮﻳﻒ اﻟﺪاﻟﺔ ‪ ln‬هﻲ‬
‫[∞‪]0; +‬‬ ‫*‪ -‬اﻟﺪاﻟﺔ ‪ ln‬ﻣﺘﺼﻠﺔ ﻋﻠﻰ‬
‫‪1‬‬
‫[ ∞‪∀ x ∈ ]0; +‬‬ ‫= ) ‪ln'( x‬‬ ‫و‬ ‫[∞‪]0; +‬‬ ‫*‪ -‬اﻟﺪاﻟﺔ ‪ ln‬ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق ﻋﻠﻰ‬
‫‪x‬‬
‫[∞‪]0; +‬‬ ‫*‪ -‬اﻟﺪاﻟﺔ ‪ ln‬ﺗﺰاﻳﺪﻳﺔ ﻗﻄﻌﺎ ﻋﻠﻰ‬
‫ﻧﺘﺎﺋﺞ‬
‫ﻟﻜﻞ ﻋﺪدﻳﻦ ﺣﻘﻴﻘﻴﻴﻦ ﻣﻮﺟﺒﻴﻦ ﻗﻄﻌﺎ ‪ x‬و ‪y‬‬
‫‪ln x = ln y ⇔ x = y‬‬
‫‪ln x ln y ⇔ x y‬‬
‫ﻣﻼﺣﻈﺔ‬
‫‪ln x = 0 ⇔ x = 1‬‬
‫‪ln x 0 ⇔ x 1‬‬
‫‪ln x ≺ 0 ⇔ 0 ≺ x ≺ 1‬‬
‫(‬
‫‪g : x → ln x 2 − 3x‬‬ ‫)‬ ‫ﺗﻤﺮﻳﻦ ‪ -1‬ﺣﺪد ﻣﺠﻤﻮﻋﺔ ﺗﻌﺮﻳﻒ اﻟﺪاﻟﺘﻴﻦ ) ‪f : x → ln ( x − 1) + ln ( 4 − x‬‬

‫(‬ ‫)‬
‫) ‪ln x 2 − 3 = ln ( 2x‬‬ ‫(‬ ‫)‬
‫اﻟﻤﻌﺎدﻟﺘﻴﻦ ‪ln x 2 + 2x = 0‬‬ ‫‪ -2‬ﺣﻞ ﻓﻲ‬

‫(‬ ‫)‬
‫) ‪ln x 2 − 2x ≤ ln ( x‬‬ ‫‪ln ( x − x − 2 ) ≺ 0‬‬
‫‪2‬‬
‫اﻟﻤﺘﺮاﺟﺤﺘﻴﻦ‬ ‫‪ -3‬ﺣﻞ ﻓﻲ‬
‫ب‪ -‬ﺧﺎﺻﻴﺔ أﺳﺎﺳﻴﺔ‬
‫) [ ∞‪∀ ( a ; b ) ∈ ( ]0; +‬‬
‫‪2‬‬
‫‪ln ( ab ) = ln a + ln b‬‬

‫‪http://arabmaths.site.voila.fr‬‬ ‫‪Moustaouli Mohamed‬‬

 ‫ ﻋﺪدﻳﻦ ﺣﻘﻴﻘﻴﻴﻦ ﻣﻮﺟﺒﻴﻦ ﻗﻄﻌﺎ‬b ‫ و‬a ‫ﻟﻴﻜﻦ‬ ‫اﻟﺒﺮهﺎن‬
F : x → ln(ax) ‫ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ‬
1 1 1
]0; +∞[ ‫ ﻋﻠﻰ‬x → ‫ داﻟﺔ أﺻﻠﻴﺔ ﻟﺪاﻟﺔ‬F ‫∀ وﻣﻨﻪ‬x ∈ ]0; +∞[ F '( x) = a ⋅ = ‫ﻟﺪﻳﻨﺎ‬
x ax x
∀ x ∈ ]0; +∞ [ F ( x ) = k + ln x ‫اذن‬
k = ln a ⇐ F (1) = k ; F (1) = ln ( a ) ⇐ x =1
∀ x ∈ ]0; +∞ [ F ( x ) = ln ( ax ) = ln a + ln x
ln ( ab ) = ln a + ln b ‫ ﻧﺤﺼﻞ ﻋﻠﻰ‬x = b ‫ﺑﻮﺿﻊ‬
‫ ﺧﺎﺻﻴﺎت‬-‫ج‬
1
∀x ∈ ]0; +∞[ ln = − ln x -*
x
x
∀ ( x ; y ) ∈ ]0; +∞[
2
ln = ln x − ln y −*
y
∀ ( x 1; x 2 ;....; x n ) ∈ ]0; +∞[ ln ( x 1 × x 2 × .......... × x n ) = ln x 1 + ln x 2 + .... + ln x n
n
−*
∀x ∈ ]0; +∞[ ∀r ∈ *
ln x r = r ln x −*

‫اﻟﺒﺮهﺎن‬

 1  1 1
ln  x ×  = ln1 ⇔ ln x + ln = 0 ⇔ ln = − ln x ™
 x  x x

ln x r = ln ( x × x × ....... × x ) = ln x + ln x + ....... + ln x = r ln x ‫ ﻓﺎن‬r ∈ *

‫إذا آﺎن‬ ™
r facteurs r termes

1
ln x r = ln x − n = ln n
= − ln x n = − n ln x = r ln x ‫ وﻣﻨﻪ‬r = −n ‫ ﻓﺈﻧﻨﺎ ﻧﻀﻊ‬r ∈ *
− ‫إذا آﺎن‬
x
p
p p
y =x q
⇔x = y q ‫ﻧﻌﻠﻢ أن‬ q∈ *
p∈ *
/ = r ‫إذا آﺎن‬
q
p p
‫اذن‬ ln y = ln x ‫ أي‬p ln x = q ln y ‫و ﺑﺎﻟﺘﺎﻟﻲ‬ ln x = ln y q ‫و ﻣﻨﻪ‬
q
p
p
ln x r = r ln x ‫ أي‬ln x q
= ln x
q
1
∀x ∈ ]0; +∞[ ln x ln x =
‫ﺣﺎﻟﺔ ﺧﺎﺻﺔ‬
2
‫ ﻣﺘﺴﺎوﻳﺘﻴﻦ ﻓﻲ اﻟﺤﺎﻟﺘﻴﻦ اﻟﺘﺎﻟﻴﺘﻴﻦ‬g ‫ و‬f ‫هﻞ اﻟﺪاﻟﺘﺎن‬ ‫ﺗﻤﺮﻳﻦ‬
f ( x ) = ln ( x − 1) g ( x ) = 2ln x − 1
2
(a
f ( x ) = ln x ( x − 1) g ( x ) = ln x + ln ( x − 1) (b
2 + 1 + ln
ln 2 −1 ‫( أﺣﺴﺐ‬1 ‫ﺗﻤﺮﻳﻦ‬
2
ln 2 0, 7 ln 3 1,1 ‫ ادا ﻋﻠﻤﺖ أن‬ln ‫ و‬ln 6 ‫( أﺣﺴﺐ ﻗﻴﻤﺔ ﻣﻘﺮﺑﺔ ﻟـ‬2
9

http://arabmaths.site.voila.fr Moustaouli Mohamed

 ‫‪ -4‬دراﺳﺔ داﻟﺔ ‪ln‬‬
‫‪ (a‬داﻟﺔ ‪ ln‬ﺗﺰاﻳﺪﻳﺔ ﻗﻄﻌﺎ ﻋﻠﻰ [∞‪]0; +‬‬
‫∞‪lim ln x = +‬‬ ‫‪ (b‬ﻣﺒﺮهﻨﺔ‪) 1‬ﻧﻘﺒﻞ(‬
‫∞‪x →+‬‬
‫∞‪lim ln x = −‬‬ ‫ﻣﺒﺮهﻨﺔ‪2‬‬
‫‪x →0 +‬‬

‫‪1‬‬ ‫‪1‬‬
‫∞‪lim+ ln x = lim ln = lim − ln t = −‬‬ ‫= ‪x‬‬ ‫ﻧﻀﻊ‬ ‫اﻟﺒﺮهﺎن‬
‫‪x →0‬‬ ‫∞‪t →+‬‬ ‫∞‪t t →+‬‬ ‫‪t‬‬
‫‪ (c‬اﻟﻌﺪد‪e‬‬
‫= ) [∞‪ ln ( ]0; +‬و ﻣﻨﻪ اﻟﺪاﻟﺔ ‪ ln‬ﺗﻘﺎﺑﻞ ﻣﻦ‬ ‫ﻟﺪﻳﻨﺎ اﻟﺪاﻟﺔ ‪ ln‬ﺗﺰاﻳﺪﻳﺔ ﻗﻄﻌﺎ ﻋﻠﻰ [∞‪ ]0; +‬وﻣﺘﺼﻠﺔ و‬
‫ﻧﺤﻮ‬ ‫[ ∞ ‪]0; +‬‬
‫و ﺑﺎﻟﺘﺎﻟﻲ اﻟﻤﻌﺎدﻟﺔ ‪ ln x = 1‬ﺗﻘﺒﻞ ﺣﻼ وﺣﻴﺪا ﻓﻲ [∞‪ ]0; +‬وﻳﺮﻣﺰ ﻟﻪ ﺑﺎﻟﺤﺮف ‪ e‬ادن ‪ln e = 1‬‬
‫‪e‬‬ ‫‪2, 71828‬‬ ‫ﻧﻘﺒﻞ أن ‪ e‬ﻟﻴﺲ ﻋﺪدا ﺟﺬرﻳﺎ و ﻗﻴﻤﺘﻪ اﻟﻤﻘﺮﺑﺔ هﻲ‬
‫‪ (d‬ﺟﺪول ﺗﻐﻴﺮات اﻟﺪاﻟﺔ ‪ln‬‬

‫ﺑﻤﺎ أن ∞‪ lim ln x = −‬ﻓﺎن ﻣﺤﻮر اﻻراﺗﻴﺐ ﻣﻘﺎرب ﻟﻠﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ اﻟﺪاﻟﺔ ‪ln‬‬ ‫‪ (e‬اﻟﻔﺮوع اﻟﻼﻧﻬﺎﺋﻴﺔ‬
‫‪x →0 +‬‬
‫‪ln x‬‬
‫‪lim‬‬ ‫‪=0‬‬ ‫ﻣﺒﺮهﻨﺔ‬
‫‪x →+∞ x‬‬
‫اذن اﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﺪاﻟﺔ ‪ ln‬ﻳﻘﺒﻞ ﻓﺮﻋﺎ ﺷﻠﺠﻤﻴﺎ ﻓﻲ اﺗﺠﺎﻩ ﻣﺤﻮر اﻷﻓﺎﺻﻴﻞ‬
‫‪1‬‬
‫اذن ﻣﻨﺤﻨﻰ اﻟﺪاﻟﺔ ‪ ln‬ﻣﻘﻌﺮ‬ ‫[∞‪∀x ∈ ]0; +‬‬ ‫‪( ln ) '' ( x ) = −‬‬ ‫‪ (f‬دراﺳﺔ اﻟﺘﻘﻌﺮ‬
‫‪x2‬‬
‫‪ (g‬اﻟﺘﻤﺜﻴﻞ اﻟﻤﺒﻴﺎﻧﻲ‬

‫‪ (h‬ﻧﻬﺎﻳﺎت هﺎﻣﺔ أﺧﺮى‬

‫ﺧﺎﺻﻴﺔ‬
‫) ‪ln (1 + x‬‬ ‫‪ln x‬‬
‫‪lim‬‬ ‫‪=1‬‬ ‫‪lim‬‬ ‫‪=1‬‬ ‫‪lim x ln x = 0‬‬
‫‪x →0‬‬ ‫‪x‬‬ ‫‪x →1 x − 1‬‬ ‫‪x →0 +‬‬

‫‪http://arabmaths.site.voila.fr‬‬ ‫‪Moustaouli Mohamed‬‬

 ‫‪x −2‬‬
‫‪x →0‬‬
‫(‬
‫‪lim− x ln x 2 − x‬‬ ‫)‬ ‫‪lim x ln ‬‬
‫∞‪x →+‬‬ ‫‪ x ‬‬
‫‪‬‬ ‫‪lim x − ln x‬‬
‫∞‪x →+‬‬
‫ﺣﺪد‬ ‫ﺗﻤﺮﻳﻦ‬

‫‪ – 5‬ﻣﺸﺘﻘﺔ اﻟﺪاﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻤﻴﺔ‬

‫أ‪ -‬ﻣﺒﺮهﻨﺔ‬
‫‪ u‬داﻟﺔ ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق ﻋﻠﻰ ﻣﺠﺎل ‪ I‬و ﻻ ﺗﻨﻌﺪم ﻋﻠﻰ هﺬا اﻟﻤﺠﺎل ‪I‬‬

‫)) ‪( ln u ( x ) ) ' = u ((x‬‬

‫‪u' x‬‬
‫‪∀x ∈ I‬‬

‫‪ u‬ﻻﺗﻨﻌﺪم ﻋﻠﻰ‪ I‬و ﻣﻨﻪ ‪ u‬إﻣﺎ ﻣﻮﺟﺒﺔ ﻗﻄﻌﺎ ﻋﻠﻰ ‪ I‬أو ﺳﺎﻟﺒﺔ ﻗﻄﻌﺎ ﻋﻠﻰ ‪I‬‬ ‫اﻟﺒﺮهﺎن‬
‫اذا آﺎﻧﺖ ‪ u‬ﻣﻮﺟﺒﺔ ﻗﻄﻌﺎ ﻋﻠﻰ ‪ I‬ﻓﺎن ) ‪f ( x ) = ln u ( x‬‬
‫)‪u '( x‬‬
‫‪∀x ∈ I‬‬ ‫= ) ‪f ' ( x ) = u ' ( x ) ln' u ( x‬‬ ‫وﻣﻨﻪ‬
‫)‪u ( x‬‬
‫اذا آﺎﻧﺖ ‪ u‬ﺳﺎﻟﺒﺔ ﻗﻄﻌﺎ ﻋﻠﻰ ‪ I‬ﻓﺎن )) ‪f ( x ) = ln(−u ( x‬‬
‫) ‪−u ' ( x‬‬ ‫)‪u '( x‬‬
‫‪∀x ∈ I‬‬ ‫= )) ‪f ' ( x ) = − u ' ( x ) ln'( − u ( x‬‬ ‫=‬ ‫وﻣﻨﻪ‬
‫) ‪−u ( x‬‬ ‫)‪u ( x‬‬
‫ﺣﺪد ﻣﺠﻤﻮﻋﺔ ﺗﻌﺮﻳﻒ اﻟﺪاﻟﺔ ‪ f‬و أﺣﺴﺐ ﻣﺸﺘﻘﺘﻬﺎ ﻓﻲ اﻟﺤﺎﻟﺘﻴﻦ اﻟﺘﺎﻟﻴﺘﻴﻦ‬ ‫ﺗﻤﺮﻳﻦ‬
‫‪f ( x ) = ln ( x 2 + 2 x ) (b‬‬ ‫‪f ( x ) = ln x 2 − 4 (a‬‬
‫ب‪ -‬ﺗﻌﺮﻳﻒ‬
‫‪ u‬داﻟﺔ ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق ﻋﻠﻰ ﻣﺠﺎل ‪ I‬و ﻻ ﺗﻨﻌﺪم ﻋﻠﻰ اﻟﻤﺠﺎل ‪I‬‬
‫'‪u‬‬
‫ﺗﺴﻤﻰ اﻟﻤﺸﺘﻘﺔ اﻟﻠﻮﻏﺎرﻳﺘﻤﻴﺔ ﻟﻠﺪاﻟﺔ ‪ u‬ﻋﻠﻰ اﻟﻤﺠﺎل ‪I‬‬ ‫اﻟﺪاﻟﺔ‬
‫‪u‬‬
‫ج‪ -‬ﻧﺘﻴﺠﺔ‬
‫‪ u‬داﻟﺔ ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق ﻋﻠﻰ ﻣﺠﺎل ‪ I‬و ﻻ ﺗﻨﻌﺪم ﻋﻠﻰ اﻟﻤﺠﺎل ‪I‬‬
‫)‪u '( x‬‬
‫→ ‪ x‬ﻋﻠﻰ ‪ I‬هﻲ اﻟﺪوال ‪ x → ln u ( x ) + c‬ﺣﻴﺚ ‪ c‬ﻋﺪد ﺛﺎﺑﺖ‬ ‫اﻟﺪوال اﻷﺻﻠﻴﺔ ﻟﺪاﻟﺔ‬
‫)‪u ( x‬‬
‫أوﺟﺪ داﻟﺔ أﺻﻠﻴﺔ ﻟﺪاﻟﺔ ‪ f‬ﻋﻠﻰ اﻟﻤﺠﺎل ‪ I‬ﻓﻲ اﻟﺤﺎﻻت اﻟﺘﺎﻟﻴﺔ‬ ‫ﺗﻤﺮﻳﻦ‪1‬‬
‫‪‬‬ ‫‪x −1‬‬ ‫) ‪f ( x ) = tan ( x‬‬ ‫‪‬‬ ‫‪x −1‬‬
‫= ) ‪f ( x‬‬ ‫‪‬‬ ‫‪f ( x ) = 2‬‬
‫‪‬‬ ‫‪x +1‬‬ ‫‪‬‬ ‫‪ −π π ‬‬ ‫‪‬‬ ‫‪x − 2x‬‬
‫[∞‪ I = ]−1; +‬‬ ‫‪‬‬ ‫‪I‬‬ ‫=‬ ‫‪ 2 ; 2 ‬‬ ‫[∞‪ I = ]2; +‬‬
‫‪‬‬ ‫‪‬‬ ‫‪‬‬
‫‪x3 + 1‬‬
‫= )‪f ( x‬‬ ‫ﺗﻤﺮﻳﻦ‪ 2‬أﺣﺴﺐ اﻟﺪاﻟﺔ اﻟﻤﺸﺘﻘﺔ ﻟﺪاﻟﺔ ‪ f‬ﻋﻠﻰ [∞‪ ]−1; +‬ﺣﻴﺚ‬
‫)‪( x + 2‬‬
‫‪2‬‬

‫‪ -II‬داﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ ﻟﻸﺳﺎس ‪a‬‬

‫‪ -1‬ﺗﻌﺮﻳﻒ‬
‫‪ a‬ﻋﺪد ﺣﻘﻴﻘﻲ ﻣﻮﺟﺐ ﻗﻄﻌﺎ و ﻣﺨﺎﻟﻒ ﻟﻠﻌﺪد ‪1‬‬
‫‪ln x‬‬
‫‪Log a‬‬ ‫→ ‪ x‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [∞‪ ]0; +‬ﺗﺴﻤﻰ داﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ ﻟﻸﺳﺎس ‪ a‬وﻧﺮﻣﺰ ﻟﻬﺎ ﺑﺎﻟﺮﻣﺰ‬ ‫اﻟﺪاﻟﺔ‬
‫‪ln a‬‬
‫‪ln x‬‬
‫[∞‪∀x ∈ ]0; +‬‬ ‫= ) ‪Log a ( x‬‬
‫‪ln a‬‬
‫ﻣﻼﺣﻈﺎت‬
‫*‪ -‬داﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ اﻟﻨﻴﺒﻴﺮي هﻲ داﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ ﻟﻸﺳﺎس ‪e‬‬
‫‪ln x‬‬
‫[∞‪∀x ∈ ]0; +‬‬ ‫= ) ‪Log e ( x‬‬ ‫‪= ln x‬‬
‫‪ln e‬‬
‫∈ ‪∀a‬‬ ‫*‪+‬‬
‫}‪− {1‬‬ ‫∈ ‪∀r‬‬ ‫‪Log a ( a ) = 1‬‬ ‫) (‬
‫*‪Log a a r = r -‬‬

‫‪http://arabmaths.site.voila.fr‬‬ ‫‪Moustaouli Mohamed‬‬

Log a ‫ ﻋﺪد ﺣﻘﻴﻘﻲ ﺛﺎﺑﺖ ﻓﺎن اﻟﺪاﻟﺔ‬k ‫ ﺣﻴﺚ‬Log a ( x ) = k ln x ]0; +∞[ ‫ ﻣﻦ‬x ‫ ﺧﺎﺻﻴﺎت ﺑﻤﺎ أن ﻟﻜﻞ‬-2
ln ‫ﺗﺤﻘﻖ ﺟﻤﻴﻊ اﻟﺨﺎﺻﻴﺎت اﻟﺘﻲ ﺗﺤﻘﻘﻬﺎ اﻟﺪاﻟﺔ‬
∀ ( x; y ) ∈ ( ]0; +∞[ )
2
∀r ∈ Log a ( xy ) = Log a ( x ) + Log a ( y )
x
( )
Log a   = Log a ( x ) − Log a ( y ) ; Log a x r = rLog a ( x )
 y

a ‫ دراﺳﺔ داﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ ﻟﻸﺳﺎس‬-3

1
∀x ∈ ]0; +∞[ Log a ' ( x ) =
x ln a
]0; +∞[ ‫ﺗﻨﺎﻗﺼﻴﺔ ﻗﻄﻌﺎ ﻋﻠﻰ‬ Log a ‫∀ اذن‬x ∈ ]0; +∞[ Log a ' ≺ 0 ‫ و ﻣﻨﻪ‬ln a ≺ 0 ‫ ﻓﺎن‬0 ≺ a ≺ 1 ‫ اذا آﺎن‬-*
lim Log a x = −∞ lim Log a x = +∞
x →+∞ x →0 +
]0; +∞[ ‫ﻗﻄﻌﺎ ﻋﻠﻰ‬ ‫ ﺗﺰاﻳﺪﻳﺔ‬Log a ‫∀ ادن‬x ∈ ]0; +∞[ Log a ' 0 ‫ و ﻣﻨﻪ‬ln a 0 ‫ ﻓﺎن‬a 1 ‫ اذا آﺎن‬-*
lim Log a x = +∞ lim Log a x = −∞
x →+∞ x →0 +

http://arabmaths.site.voila.fr Moustaouli Mohamed

 ‫‪ -4‬ﺣﺎﻟﺔ ﺧﺎﺻﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ اﻟﻌﺸﺮي‬
‫ﺗﻌﺮﻳﻒ‬
‫اﻟﺪاﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻤﻴﺔ اﻟﺘﻲ أﺳﺎﺳﻬﺎ ‪ 10‬ﺗﺴﻤﻰ داﻟﺔ اﻟﻠﻮﻏﺎرﻳﺘﻢ اﻟﻌﺸﺮي و ﻳﺮﻣﺰ ﻟﻬﺎ ﺑـ ‪log‬‬
‫‪ln x‬‬
‫[∞‪∀x ∈ ]0; +‬‬ ‫= ‪log x = Log 10 x‬‬
‫‪ln10‬‬

‫ﻣﻼﺣﻈﺎت‬
‫‪1‬‬
‫= ‪ M‬ﻓﺎﻧﻨﺎ ﻧﺤﺼﻞ ﻋﻠﻰ‬ ‫*‪ -‬اذا وﺿﻌﻨﺎ‬
‫‪ln10‬‬
‫‪(M‬‬ ‫) ‪0, 434‬‬ ‫[∞‪∀x ∈ ]0; +‬‬ ‫‪log x = M ln x‬‬
‫∈ ‪∀m‬‬ ‫‪log10m = m‬‬ ‫*‪-‬‬
‫‪log 0, 01‬‬ ‫ﺗﻤﺮﻳﻦ ‪ -1‬أﺣﺴﺐ ‪log10000‬‬
‫‪log ( x − 1) + log ( x + 3) = 2‬‬ ‫‪ -2‬ﺣﻞ ﻓﻲ‬
‫‪ x + y = 65‬‬
‫‪‬‬
‫‪2‬‬
‫‪ -3‬ﺣﻞ ﻓﻲ‬
‫‪log x + log y = 3‬‬

‫‪http://arabmaths.site.voila.fr‬‬ ‫‪Moustaouli Mohamed‬‬