08 08 teoremi logaritmi 3 0

Logaritmi: teoremi
Logaritmi
applicando i teoremi sui logaritmi riduci ad un unico logaritmo
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v 3.0
𝑙𝑙𝑙𝑙𝑙𝑙 2 + 𝑙𝑙𝑙𝑙𝑙𝑙 3
𝑙𝑙𝑙𝑙𝑙𝑙 6
𝑙𝑙𝑙𝑙𝑙𝑙 3 + 𝑙𝑙𝑙𝑙𝑙𝑙 4 − 𝑙𝑙𝑙𝑙𝑙𝑙 2
𝑙𝑙𝑙𝑙𝑙𝑙 6
𝑙𝑙𝑙𝑙𝑙𝑙2 4 − 𝑙𝑙𝑙𝑙𝑙𝑙2 2
1
2
𝑙𝑙𝑙𝑙𝑙𝑙1 3 − �𝑙𝑙𝑙𝑙𝑙𝑙1 + 𝑙𝑙𝑙𝑙𝑙𝑙1 3�
3
33
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3
32
𝑙𝑙𝑙𝑙𝑙𝑙1
𝑙𝑙𝑙𝑙𝑙𝑙 4 − (𝑙𝑙𝑙𝑙𝑙𝑙 7 + 𝑙𝑙𝑙𝑙𝑙𝑙 2) + 𝑙𝑙𝑙𝑙𝑙𝑙 14
𝑙𝑙𝑙𝑙𝑙𝑙 4
1
𝑙𝑙𝑙𝑙𝑙𝑙2 5
2
𝑙𝑙𝑙𝑙𝑙𝑙2 √5
2 𝑙𝑙𝑙𝑙𝑙𝑙 3
− 𝑙𝑙𝑙𝑙𝑙𝑙1 7
2
2 𝑙𝑙𝑙𝑙𝑙𝑙 3 + 𝑙𝑙𝑙𝑙𝑙𝑙
𝑙𝑙𝑙𝑙𝑙𝑙 9
1
27
𝑙𝑙𝑙𝑙𝑙𝑙1
1
3
𝑙𝑙𝑙𝑙𝑙𝑙 3
2
𝑙𝑙𝑙𝑙𝑙𝑙 7 − 𝑙𝑙𝑙𝑙𝑙𝑙 49
3
2 𝑙𝑙𝑙𝑙𝑙𝑙3 𝑎𝑎 + 2 𝑙𝑙𝑙𝑙𝑙𝑙3 𝑏𝑏 − 𝑙𝑙𝑙𝑙𝑙𝑙3 𝑎𝑎 − 𝑙𝑙𝑙𝑙𝑙𝑙3 𝑏𝑏
𝑙𝑙𝑙𝑙𝑙𝑙 𝑚𝑚 − 𝑙𝑙𝑙𝑙𝑙𝑙 𝑛𝑛 + 𝑙𝑙𝑙𝑙𝑙𝑙(𝑚𝑚 + 𝑛𝑛)
3 𝑙𝑙𝑙𝑙𝑙𝑙 𝑝𝑝 + 4 𝑙𝑙𝑙𝑙𝑙𝑙 𝑞𝑞 − 2 𝑙𝑙𝑙𝑙𝑙𝑙 𝑟𝑟
4
− 𝑙𝑙𝑙𝑙𝑙𝑙 7
3
𝑙𝑙𝑙𝑙𝑙𝑙3 (𝑎𝑎𝑎𝑎)
𝑙𝑙𝑙𝑙𝑙𝑙
𝑙𝑙𝑙𝑙𝑙𝑙
2 𝑙𝑙𝑙𝑙𝑙𝑙0,1 𝑎𝑎 + 𝑙𝑙𝑙𝑙𝑙𝑙0,1 𝑏𝑏 − 4 𝑙𝑙𝑙𝑙𝑙𝑙0,1 𝑐𝑐
𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎2 − 𝑏𝑏 2 ) − 𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 + 𝑏𝑏) + 𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 − 𝑏𝑏)
1
3
𝑙𝑙𝑙𝑙𝑙𝑙(𝑚𝑚 + 𝑛𝑛) + 2 𝑙𝑙𝑙𝑙𝑙𝑙(𝑚𝑚 + 𝑛𝑛) − 𝑙𝑙𝑙𝑙𝑙𝑙(𝑚𝑚 + 𝑛𝑛)
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2
𝑝𝑝 𝑞𝑞
𝑙𝑙𝑙𝑙𝑙𝑙 𝑝𝑝𝑝𝑝 + 𝑙𝑙𝑙𝑙𝑙𝑙 � − �
𝑞𝑞 𝑝𝑝
𝑚𝑚
− 1� + 𝑙𝑙𝑙𝑙𝑙𝑙 𝑛𝑛 − 𝑙𝑙𝑙𝑙𝑙𝑙(𝑚𝑚 − 𝑛𝑛)
𝑛𝑛
𝑙𝑙𝑙𝑙𝑙𝑙 �
𝑚𝑚(𝑚𝑚 + 𝑛𝑛)
𝑛𝑛
𝑝𝑝3 𝑞𝑞 4
𝑟𝑟 2
𝑙𝑙𝑙𝑙𝑙𝑙0,1
𝑎𝑎2 𝑏𝑏
𝑐𝑐 4
𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 − 𝑏𝑏)2
𝑙𝑙𝑙𝑙𝑙𝑙(𝑚𝑚 + 𝑛𝑛)
𝑙𝑙𝑙𝑙𝑙𝑙(𝑝𝑝2 − 𝑞𝑞 2 )
0
1
1
1
𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎 + 𝑙𝑙𝑙𝑙𝑙𝑙 𝑏𝑏 + 𝑙𝑙𝑙𝑙𝑙𝑙 𝑐𝑐
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4
8
8
𝑙𝑙𝑙𝑙𝑙𝑙 �𝑎𝑎4 𝑏𝑏 2 𝑐𝑐
2 𝑙𝑙𝑙𝑙𝑙𝑙 𝑝𝑝 + 5 𝑙𝑙𝑙𝑙𝑙𝑙 𝑞𝑞 − (3 𝑙𝑙𝑙𝑙𝑙𝑙 𝑟𝑟 − 4 𝑙𝑙𝑙𝑙𝑙𝑙 𝑝𝑝)
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𝑙𝑙𝑙𝑙𝑙𝑙
𝑝𝑝6 𝑞𝑞 5
𝑟𝑟 3
1 di 3
Logaritmi: teoremi
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applicando i teoremi sui logaritmi sviluppa le seguenti espressioni
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𝑙𝑙𝑙𝑙𝑙𝑙 10
𝑙𝑙𝑙𝑙𝑙𝑙 5 + 𝑙𝑙𝑙𝑙𝑙𝑙 2
𝑙𝑙𝑙𝑙𝑙𝑙1 √8
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𝑙𝑙𝑙𝑙𝑙𝑙1 2
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𝑙𝑙𝑙𝑙𝑙𝑙3
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24
𝑙𝑙𝑙𝑙𝑙𝑙 �2
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𝑙𝑙𝑙𝑙𝑙𝑙4 �
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v 3.0
𝑙𝑙𝑙𝑙𝑙𝑙3 7 − 𝑙𝑙𝑙𝑙𝑙𝑙3 2
3 5
𝑙𝑙𝑙𝑙𝑙𝑙 �
7
1 − 𝑎𝑎
�
𝑎𝑎2 𝑏𝑏
𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎2 − 2𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 )
𝑙𝑙𝑙𝑙𝑙𝑙5 (𝑎𝑎2 − 𝑏𝑏 2 )
𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎3 − 𝑏𝑏 3 )
𝑙𝑙𝑙𝑙𝑙𝑙 �
𝑎𝑎2 − 𝑏𝑏 2
�
𝑎𝑎𝑎𝑎
1
(𝑙𝑙𝑙𝑙𝑙𝑙 2 + 𝑙𝑙𝑙𝑙𝑙𝑙 3 − 𝑙𝑙𝑙𝑙𝑙𝑙 5)
2
𝑙𝑙𝑙𝑙𝑙𝑙4 (1 − 𝑎𝑎) − 2 𝑙𝑙𝑙𝑙𝑙𝑙4 𝑎𝑎 − 𝑙𝑙𝑙𝑙𝑙𝑙4 𝑏𝑏
1
1
𝑙𝑙𝑙𝑙𝑙𝑙 5 − 𝑙𝑙𝑙𝑙𝑙𝑙 7
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2 𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 − 𝑏𝑏)
𝑙𝑙𝑙𝑙𝑙𝑙5 (𝑎𝑎 + 𝑏𝑏) + 𝑙𝑙𝑙𝑙𝑙𝑙5 (𝑎𝑎 − 𝑏𝑏)
𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 − 𝑏𝑏) + 𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎2 + 𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 )
𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 − 𝑏𝑏) + 𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 + 𝑏𝑏) − 𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎 − 𝑙𝑙𝑙𝑙𝑙𝑙 𝑏𝑏
𝑙𝑙𝑙𝑙𝑙𝑙 �𝑎𝑎√𝑏𝑏
1
1
𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎 + 𝑙𝑙𝑙𝑙𝑙𝑙 𝑏𝑏
2
4
𝑙𝑙𝑙𝑙𝑙𝑙(𝑝𝑝7 𝑞𝑞)4
28 𝑙𝑙𝑙𝑙𝑙𝑙 𝑝𝑝 + 4 𝑙𝑙𝑙𝑙𝑙𝑙 𝑞𝑞
𝑙𝑙𝑙𝑙𝑙𝑙 𝑚𝑚2 √𝑛𝑛
1
𝑛𝑛3 3
𝑙𝑙𝑙𝑙𝑙𝑙 � 2 �
𝑚𝑚
𝑙𝑙𝑙𝑙𝑙𝑙 �
𝑧𝑧
2 2
𝑥𝑥 𝑦𝑦
3
2
�
4
𝑙𝑙𝑙𝑙𝑙𝑙6 �𝑎𝑎2𝑛𝑛+1
𝑙𝑙𝑙𝑙𝑙𝑙
𝑥𝑥(𝑥𝑥 + 𝑦𝑦)
𝑧𝑧
1
𝑙𝑙𝑙𝑙𝑙𝑙 4
√𝑎𝑎3 𝑏𝑏 2 𝑐𝑐
𝑙𝑙𝑙𝑙𝑙𝑙5
4
�(𝑥𝑥 − 𝑦𝑦)3
5
��𝑥𝑥 + 𝑦𝑦
1 𝑎𝑎 − 1 𝑎𝑎 + 2 2
𝑙𝑙𝑙𝑙𝑙𝑙 � �
�
� �
3 𝑎𝑎 + 1 𝑎𝑎 − 2
1
2 𝑙𝑙𝑙𝑙𝑙𝑙 𝑚𝑚 + 𝑙𝑙𝑙𝑙𝑙𝑙 𝑛𝑛
2
2
𝑙𝑙𝑙𝑙𝑙𝑙 𝑛𝑛 − 𝑙𝑙𝑙𝑙𝑙𝑙 𝑚𝑚
3
2
2 𝑙𝑙𝑙𝑙𝑙𝑙 𝑧𝑧 − 2 𝑙𝑙𝑙𝑙𝑙𝑙 − 4 𝑙𝑙𝑙𝑙𝑙𝑙 𝑥𝑥 − 2 𝑙𝑙𝑙𝑙𝑙𝑙 𝑦𝑦
3
2𝑛𝑛 + 1
𝑙𝑙𝑙𝑙𝑙𝑙6 𝑎𝑎
4
𝑙𝑙𝑙𝑙𝑙𝑙 𝑥𝑥 + 𝑙𝑙𝑙𝑙𝑙𝑙(𝑥𝑥 + 𝑦𝑦) − 𝑙𝑙𝑙𝑙𝑙𝑙 𝑧𝑧
3
1
1
− 𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎 − 𝑙𝑙𝑙𝑙𝑙𝑙 𝑏𝑏 − 𝑙𝑙𝑙𝑙𝑙𝑙 𝑐𝑐
4
2
4
3
1
𝑙𝑙𝑙𝑙𝑙𝑙5 (𝑥𝑥 − 𝑦𝑦) −
𝑙𝑙𝑙𝑙𝑙𝑙5 (𝑥𝑥 + 𝑦𝑦)
4
10
1
1
1
𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 − 1) − 𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 + 1) + 𝑙𝑙𝑙𝑙𝑙𝑙 +
2
2
3
+2 𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 + 2) − 2 𝑙𝑙𝑙𝑙𝑙𝑙(𝑎𝑎 − 2)
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Logaritmi: teoremi
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esercizi di riepilogo
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v 3.0
2𝑙𝑙𝑙𝑙 3 + 𝑙𝑙𝑙𝑙 4
𝑙𝑙𝑙𝑙1 − 𝑙𝑙𝑙𝑙
𝑙𝑙𝑙𝑙 36
1
3
𝑙𝑙𝑙𝑙16 + 𝑙𝑙𝑙𝑙
𝑙𝑙𝑙𝑙3
1
16
0
𝑙𝑙𝑙𝑙3 8 − 𝑙𝑙𝑙𝑙9 16
𝑙𝑙𝑙𝑙3 2
𝑙𝑙𝑙𝑙(𝑎𝑎2 − 𝑏𝑏 2 ) − 𝑙𝑙𝑙𝑙(𝑎𝑎 + 𝑏𝑏)
𝑙𝑙𝑙𝑙(𝑎𝑎 − 𝑏𝑏)
5𝑙𝑙𝑙𝑙 3 + 8𝑙𝑙𝑙𝑙 4
𝑙𝑙𝑙𝑙(35 ∙ 48 )
𝑙𝑙𝑙𝑙(𝑎𝑎 + 𝑏𝑏) − 𝑙𝑙𝑙𝑙𝑙𝑙 − 𝑙𝑙𝑙𝑙𝑙𝑙
𝑙𝑙𝑙𝑙 �
𝑎𝑎
� − 𝑙𝑙𝑙𝑙𝑙𝑙
3𝑏𝑏
𝑎𝑎 + 𝑏𝑏
�
𝑎𝑎 𝑏𝑏
1
3𝑏𝑏
𝑙𝑙𝑙𝑙 �
𝑙𝑙𝑙𝑙
𝑙𝑙𝑙𝑙 7𝑎𝑎 + 𝑙𝑙𝑙𝑙 3𝑏𝑏 − 𝑙𝑙𝑙𝑙 9𝑏𝑏 2 + 𝑙𝑙𝑙𝑙 𝑎𝑎
𝑙𝑙𝑙𝑙 �
𝑙𝑙𝑙𝑙(𝑎𝑎 + 𝑏𝑏)2 − 𝑙𝑙𝑙𝑙𝑎𝑎2
𝑏𝑏 2
𝑙𝑙𝑙𝑙 �1 + �
𝑎𝑎
7𝑎𝑎2
�
3𝑏𝑏
3 3
𝑙𝑙𝑙𝑙 �
2
1
(𝑙𝑙𝑙𝑙3 − 𝑙𝑙𝑙𝑙2)
3
𝑙𝑙𝑙𝑙(𝑎𝑎3 ∙ 𝑏𝑏 3 )
3(𝑙𝑙𝑙𝑙𝑙𝑙 + 𝑙𝑙𝑙𝑙 𝑏𝑏)
𝑎𝑎2 + 𝑏𝑏 2
𝑙𝑙𝑙𝑙 �
�
5𝑎𝑎
𝑙𝑙𝑙𝑙(𝑎𝑎2 + 𝑏𝑏 2 ) − 𝑙𝑙𝑙𝑙5𝑎𝑎
𝑎𝑎2 + 𝑏𝑏 2 + 2𝑎𝑎𝑎𝑎
𝑙𝑙𝑙𝑙�
𝑎𝑎 + 𝑏𝑏
𝑙𝑙𝑙𝑙 �
3𝑎𝑎 + 5𝑏𝑏
�
𝑎𝑎(3𝑎𝑎 − 5𝑏𝑏)
𝑙𝑙𝑙𝑙 �
5
�
7𝑎𝑎 + 4𝑏𝑏
𝑙𝑙𝑙𝑙 �
1
𝑙𝑙𝑙𝑙(𝑎𝑎 + 𝑏𝑏)
2
𝑎𝑎3 − 𝑏𝑏 3
�
𝑎𝑎2 + 𝑏𝑏 2 + 𝑎𝑎𝑎𝑎
𝑙𝑙𝑙𝑙 �
15
�√3 𝑎𝑎 + √3 𝑏𝑏�
2�
𝑙𝑙𝑙𝑙(3𝑎𝑎 + 5𝑏𝑏) − 𝑙𝑙𝑙𝑙𝑙𝑙 − 𝑙𝑙𝑙𝑙(3𝑎𝑎 − 5𝑏𝑏)
𝑙𝑙𝑙𝑙(𝑎𝑎 − 𝑏𝑏)
𝑙𝑙𝑙𝑙5 − 𝑙𝑙𝑙𝑙(7𝑎𝑎 + 4𝑏𝑏)
𝑙𝑙𝑙𝑙 5 − 2 𝑙𝑙𝑙𝑙(𝑎𝑎 + 𝑏𝑏)
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