מתמטיקה - 4 יח"ל - כיתה י"ב - חלק א'
ﺟﺑﺭ ﻭﻣﻘﺩّﻣﺔ ﻟﻠﺗّﺣﻠﻳﻝ ﺍﻟﺭّﻳﺎﺿﻲ ﻟﻠﺩّﻭﺍﻝ ﺍﻷﺳّﻳّﺔ ﻭﺍﻟﻠّﻭﻏﺭﻳﺛﻣﻳّﺔ
- 59 -
ﺍﻟﻔﺻﻝ :5 ﺍﻟﻠّﻭﻏﺭﻳﺛﻣﺎﺕ
log b x x
ﻧﻌﺮﺽ ﻓﻴﻤﺎ ﻳﻠﻲ ﺇﺟﻤﺎﻝ ﻟﻘﻮﺍﻧﻴﻦ ﺍﻟﻠّﻮﻏﺮﻳﺜﻤﺎﺕ، ﺍﻟﻨّﺎﺗﺠﺔ ﻋﻦ ﺗﻌﺮﻳﻒ ﺍﻟﻠّﻮﻏﺮﻳﺜﻢ
b
a
a
log
x
x
) 1 (
a
a
1 a a a a
a log a
ﻓﻲ ﺟﻤﯿﻊ اﻟﺨﺼﺎﺋﺺ اﻟﺜّﻼث ، a ﻣﻮﺟﺐ
ﻷنّ
) 2 (
1
a .
a , 1
b a log a b a 1 0 log
b b
وﻻ ﻳﺴﺎوي 1 ، أي : 0
ﻷنّ
) 3 (
0 a
ﻷنّ
) 4 (
1
) 5 ( ﻟﻮﻏﺮﻳﺜﻢ ﺣﺎﺻﻞ ﺿﺮب
0 y x , , 0 a
a a a log ( ) log log xy x y
1
0
) 6 ( ﻟﻮﻏﺮﻳﺜﻢ ﻗﺴﻤﺔ
a log log _ i x y x
0 y x , , 0 a
log
1
0
y
a
a
) 7 ( ﻟﻮﻏﺮﻳﺜﻢ ﻗﻮّة
b log
0 a x ,
b
log
1
0
x
x
a
a
ﺗﻤﺎﺭﻳﻦ ﻟﻠﻌﻤﻞ ﺍﻟﺬّﺍﺗﻲ
(1) ﺍِﺣﺴﺒﻮﺍ ﻗﻴﻢ ﺍﻟ ﻠّﻮﻏﺮﻳﺜﻤﺎﺕ ﺍﻟﺘّﺎﻟﻴﺔ .
1 4 9
1 6
)ﺩ(
)ﺝ(
)ﺏ(
)ﺃ(
8 1 log
4 16 log
log
log
6
7
1
1 3
)ﺡ(
)ﺯ(
)ﻭ(
)ﻫـ(
log
log
log
log
8
4
8
7
3
2
7
log
x
x
ﺣﺴﺐ ﺗﻌﺮﻳﻒ ﺍﻟﻠّﻮﻏﺮﻳﺜﻢ، ﻳﺘﺤﻘّﻖ :
(2) ﺗﺬﻛﯿﺮ:
a
.
a
3 a k 2 log 2
4
log
log
log
4
25
3
4
3
2 5
5
16
2
3 81
5
2
2
ﻣﺜﺎل:
ﺍِﺣﺴﺒﻮﺍ ﻗﻴﻢ ﺍﻟﺘّﻌﺎﺑﻴﺮ ﺍﻟﺘّﺎﻟﻴﺔ ﺩﻭﻥ ﺍِﺳﺘﻌﻤﺎﻝ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ .
log ﺑﺪﻭﻥ ﻛﺘﺎﺑﺔ ﺃﺳﺎﺱ ﻳﻤﺜّﻞ ﻟﻮﻏﺮﻳﺜﻢ ﺣﺴﺐ ﺍﻷﺳﺎﺱ . 10 ln ﻳﻤﺜّﻞ ﻟﻮﻏﺮﻳﺜﻢ ﺣﺴﺐ ﺍﻷﺳﺎﺱ e .
ﻧﺬﻛّﺮﻛﻢ :
log
log
log
log
3
1
1
4
2
4
1
5
)ﺩ(
)ﺝ(
)ﺏ(
)ﺃ(
8
3
2
5
8
3
5
2
5 log 3 _ i 3
log
log
2
2
ln
1 6
5
e
)ﺡ(
)ﺯ(
)ﻭ(
)ﻫـ(
4
4
e
e
e
log
ln
5 ln
7 ln
2 5 2
3
1
)ﻱ ﺏ(
)ﻱ ﺃ(
)ﻱ(
)ﻁ(
10
e
e
e
2 e _ i
3 ln
5 ln
2 2 ln
ln3 ln4
2
) ﻁ ﺯ (
) ﻁ ﻭ (
) ﻱ ﺩ (
)ﻱ ﺝ(
e
e
e
© ﺟﻣﻳﻊ ﺍﻟﺣﻘﻭﻕ ﻣﺣﻔﻭﻅﺔ ﻟﺟﺎﺑﻲ ﻳﻛﻭﺋﻳﻝ –
– ﺭﻳﺎﺿﻳّﺎﺕ ﻟﻁﻼّﺏ 4 ﻭﺣﺩﺍﺕ ﺗﻌﻠﻳﻣﻳّﺔ – ﺍﻟﺻّﻑ ﺍﻟﺛّﺎﻧﻲ ﻋﺷﺭ – ﺍﻟﻣﻧﻬﺞ ﺍﻟﺟﺩﻳﺩ
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