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´ Ãåíéêïý Ëõêåßïõ - Åõðáëßïõ

ÊáèçãçôÞò: ÌÁÍÙËÇÓ Êùíóôáíôßíïò

5.2  ËïãÜñéèìïé

ÅéóáãùãÞ

Áí  θ  åßíáé Ýíáò èåôéêüò áñéèìüò êáé   0 < α ≠ 1 ôüôå ç åîßóùóç   α x = θ   Ý÷åé ìïíáäéêÞ ëýóç (ãéáôß ç åêèåôéêÞ óõíÜñôçóç   f   ìå   f ⁡ ( x ) = α x   åßíáé  " "1-1" "  êáé ï  θ  áíÞêåé óôï óýíïëï ôéìþí ôçò).

Ôç ìïíáäéêÞ áõôÞ ëýóç ôç óõìâïëßæïõìå   log α ⁢ θ   êáé ôçí ïíïìÜæïõìå ëïãÜñéèìï ôïõ  θ  ìå âÜóç  α.

¸ôóé Ý÷ïõìå ôïí ïñéóìü ëïãáñßèìïõ:

ÏíïìÜæïõìå ëïãÜñéèìï ôïõ èåôéêïý áñéèìïý  θ  ìå âÜóç  α   ( 0 < α ≠ 1 )   ôïí åêèÝôç óôïí ïðïßï ðñÝðåé íá õøþóïõìå ôïí  α  ãéá íá âñïýìå ôï  θ.

¢ñá:

α x = θ ⇔ x = log α ⁢ θ

ÐáñÜäåéãìá

log 2 ⁢ 32 = 5 ⁢ ãéáôß ⁢ 2 5 = 32

ÐáñáôçñÞóåéò:

Áí   0 < α ≠ 1 , θ > 0   êáé   x ∈   ôüôå áðü ôïí ïñéóìü ôïõ ëïãáñßèìïõ ðñïêýðôïõí ïé ó÷Ýóåéò:

log α ⁢ α x = x ⁢ α log α ⁢ θ = θ log α ⁢ α = 1 ⁢ log α ⁢ 1 = 0 θ 1 = θ 2 ⇔ log α ⁢ θ 1 = log α ⁢ θ 2

Éäéüôçôåò ôùí ëïãáñßèìùí

Áí   0 < α ≠ 1 ,   ôüôå ãéá ïðïéïõóäÞðïôå èåôéêïýò áñéèìïýò   θ 1 , θ 2 , θ   êáé   κ ∈   éó÷ýïõí:

1. log α ⁡ ( θ 1 ⁢ θ 2 ) = log α ⁢ θ 1 + log α ⁢ θ 2

2.   log α ⁢ θ 1 θ 2 = log α ⁢ θ 1 - log α ⁢ θ 2

3.   log α ⁢ θ κ = κ ⁢ log α ⁢ θ

Ðüñéóìá

Áí   0 < α ≠ 1 , θ > 0   êáé   ν ∈ *   ôüôå:

log α ⁢ θ ν = 1 ν ⁢ log α ⁢ θ
Óçìåßùóç

Ïé ÷ñçóéìïðïéïýìåíåò âÜóåéò ôùí ëïãáñßèìùí åßíáé óõíÞèùò ôï  10  êáé ôï   ⅇ .

Ïé ëïãÜñéèìïé ìå âÜóç ôï  10  ëÝãïíôáé äåêáäéêïß Þ  êïéíïß ëïãÜñéèìïé êáé óõìâïëßæïíôáé ìå   logθ   áíôß ôïõ   log 10 ⁢ θ .
¢ñá:  

logθ = x ⇔ 10 x = θ

Ïé ëïãÜñéèìïé ìå âÜóç ôï  ⅇ  ëÝãïíôáé öõóéêïß  Þ  íåðÝñéïé ëïãÜñéèìïé êáé óõìâïëßæïíôáé ìå   lnθ   áíôß ôïõ   log ⅇ ⁢ θ .
¢ñá:  

lnθ = x ⇔ ⅇ x = θ
Ðñüôáóç (Ôýðïò áëëáãÞò âÜóçò)

Áí   0 < α ≠ 1   êáé   0 < β ≠ 1 ,   ôüôå ãéá êÜèå èåôéêü áñéèìü  θ  éó÷ýåé:

log β ⁢ θ = log α ⁢ θ log α ⁢ β

Áðü ôïí ðáñáðÜíù ôýðï ðñïêýðôïõí ôá åîÞò:

•  Ç áëëáãÞ ôçò âÜóçò óôïõò äåêáäéêïýò ëïãÜñéèìïõò äßíåôáé áðü ôïí ôýðï:

log β ⁢ θ = log ⁢ θ log ⁢ β

•  Ç áëëáãÞ ôçò âÜóçò óôïõò öõóéêïýò ëïãÜñéèìïõò äßíåôáé áðü ôïí ôýðï:

log β ⁢ θ = ln ⁢ θ ln ⁢ β

ÁóêÞóåéò

0.  Íá õðïëïãéóôïýí ïé ëïãÜñéèìïé:

á)   log 4 ⁢ 16                    â)   log 1 2 ⁢ 32                                   ã)   log 2 ⁢ 1 8
0.  Íá õðïëïãéóôïýí ïé ëïãÜñéèìïé:

á)   log 3 ⁢ 1 3                                       â)   log 1 2 ⁢ 64                                   ã)   log 9 ⁢ 3
0.  Íá õðïëïãéóôïýí ïé ëïãÜñéèìïé:

á)   log 9 ⁢ 27                                    â)   log 3 ⁢ 1 27                                   ã)   log 4 ⁢ 2
0.  Íá õðïëïãéóôïýí ïé ëïãÜñéèìïé:

á)   log 4 ⁢ 32                                       â)   log 0.1 ⁢ 100                                ã)   log 8 ⁢ 2 4
0.  Íá õðïëïãéóôïýí ïé ëïãÜñéèìïé:

á)   log 2 ⁢ 1 2                                       â)   log 5 ⁢ 125                                   ã)   log α ⁢ α 3
0.  Íá õðïëïãéóôïýí ïé ëïãÜñéèìïé:

á)   log 1 9 ⁢ 3 3                                  â)   log 0.2 ⁢ 625                                ã)   log ⁢ 10 ⁢ 10
0.  Áí   log 3 ⁢ 2 = α ,   íá õðïëïãßóåôå ôïí log 8 ⁢ 12.
0.  Íá âñåèåß ï ðñáãìáôéêüò áñéèìüò   x   áí:

á)   log 9 ⁢ x = 1 2                                â)   log x ⁢ 27 = 3                              ã)   log x ⁢ 5 = 1 4
0.  Íá âñåèåß ï ðñáãìáôéêüò áñéèìüò   x   áí:

á)   log x ⁢ 4 = 2                                  â)   log x ⁢ 8 = 3                            ã)   log 1 2 ⁢ x = 2
0.  Íá âñåèåß ï ðñáãìáôéêüò áñéèìüò   x   áí:

á)   log 3 ⁢ x = 4                                  â)   log ⁢ x = - 1                                ã)   ln ⁢ x = 2
0.  Íá âñåèåß ï ðñáãìáôéêüò áñéèìüò   x   áí:

á)   log ⁢ x = 1 3                                  â)   log 27 ⁢ x = 1 3                            ã)   ln ⁢ x = ⅇ
0.  Íá âñåèåß ï ðñáãìáôéêüò áñéèìüò   x   áí:

á)   log 2 ⁢ x = 6                              â)   log 3 ⁢ x 2 = 4                        ã)   log 4 9 ( x - 1 x ) = - 1 2
0.  Íá âñåèåß ï ðñáãìáôéêüò áñéèìüò   x   áí:

á)   log x ⁢ 1000 = - 6                          â)   log x ⁢ 16 = 2 3                          ã)   log x ⁢ 16 81 = 4
0.  Íá âñåèåß ï ðñáãìáôéêüò áñéèìüò   x   áðü ôéò ðáñáêÜôù ó÷Ýóåéò:

á)   log 1 9 ⁢ 3 3 = x                            â)   log 0.1 ⁢ 100 5 = x                   ã)   log x ⁢ 27 = 3 2
0.  Íá âñåèåß ï ðñáãìáôéêüò áñéèìüò   x   áðü ôéò ðáñáêÜôù ó÷Ýóåéò:

á)   log x ⁢ 4 = - 3 2                              â)   log 8 ⁢ x = - 1 3                          ã)   log 4 ⁡ ( log x ⁢ 25 ) = 1 2
0.  Íá õðïëïãéóôåß ï   x   üôáí éó÷ýåé:

á)   5 x = 5                     â)   ln ⁢ x = 3                          ã)   log x ⁢ 4 = x                      ä)   log x ⁢ 2 ⁢ x = 2
0.  Íá åöáñìüóåôå üëåò ôéò éäéüôçôåò ôùí ëïãáñßèìùí:

á)   log 3 ⁢ 3 ⁢ α 6 ⁢ β γ                              â)   log ⁢ 10 ⁢ α 2 ⁢ β 3 3 ⁢ β 2                          ã)   log 100 ⁢ 10 ⁢ α 2 ⁢ β
0.  Íá âñåèåß ï áêÝñáéïò áñéèìüò   x   Ýôóé þóôå íá Ý÷ïõí Ýííïéá óôï  χ΅  ôá óýìâïëá:

á)   log x ⁡ ( 2 - | x | )                             â)   log x ⁢ 1 + x 5 - x                                 ã)   log 2 ⁢ x ⁢ x + 1 3 - x
0.  Äåßîåôå üôé:

á)   2 ⁢ log2 + 3 ⁢ log3 - log12 = 2 ⁢ log3          

â)   1 2 ⁢ log16 + 1 3 ⁢ log8 + 1 4 ⁢ log81 = 3 ⁢ log2 + log3
0.  Íá áðïäåßîåôå üôé:

á)   3 ⁢ log 3 ⁢ 2 + 2 ⁢ log 3 ⁢ 6 - log 3 ⁢ 32 = 2                                â)   2 + 3 ⁢ log 5 ⁢ 2 - 2 ⁢ log 5 ⁢ 10 = log 5 ⁢ 2
0.  Äåßîåôå üôé:   

log 6 ⁡ ( 12 + 6 ⁢ 3 ) + 2 ⁢ log ⁡ ( 3 - 3 ) = 2          
0.  Äåßîåôå üôé:    log2 + log ⁡ ( 3 + 1 ) + log ⁡ ( 1 + 2 - 3 ) + log ⁡ ( 1 - 2 - 3 ) = 2 ⁢ log2          
0.  Äåßîåôå üôé:    1 + log 3 ⁢ 5 - 2 ⁢ log 3 ⁢ 10 4 = 1 + log 3 ⁢ 2          
0.  Äåßîôå üôé:    3 log 3 ⁢ 8 - log 3 ⁢ 2 = 4
0.  Íá õðïëïãéóôåß ç ðáñÜóôáóç:    Á = log 2006 [ log 5 ⁡ ( log 2 ⁢ 32 ) ]
0.  Íá õðïëïãßóåôå ôçí ðáñÜóôáóç:    Á = 100 1 2 - log ⁢ 4 4
0.  Íá áðïäåßîåôå üôé:

á)   10 3 ⁢ log2 + log5 - 1 = 4                                                    â)   100 1 - 1 4 ⁢ log25 = 20
0.  Áí   log 2 ⁢ x = α , log 2 ⁢ y = β    êáé    log 2 ⁢ ω = γ   íá õðïëïãßóåôå ôïõò:

á)   log 2 ⁢ x 2 ⁢ y 4 ⁢ ω                                                             â)   log 2 ⁢ x ⁢ y 3

ã)   log 4 ( 32 ⁢ x 3 ⁢ y 2 ⁢ ω )                                                        ä)   log 1 2 ⁡ ( ( x + y ) 2 - ( x - y ) 2 )
0.  Áí   log2 = α , log3 = β    íá âñåèïýí ïé ëïãÜñéèìïé ôùí áñéèìþí:      4 , 5 , 6 , 12 , 15 , 30 , 36 , 72 50 .
0.  Íá áðïäåßîåôå üôé:

á)   1 2 ⁢ log25 + 1 3 ⁢ log8 + 1 5 ⁢ log32 = 1 + log2

â)   log 2 ⁢ 3 · log 3 ⁢ 4 · log 4 ⁢ 5 · log 5 ⁢ 6 · log 6 ⁢ 7 · log 7 ⁢ 8 = 3
0.  Äåßîôå üôé:

á)   7 16 ⁢ log ⁡ ( 3 + 2 ⁢ 2 ) - 4 ⁢ log ⁡ ( 2 + 1 ) = 25 8 ⁢ log ( 2 - 1 )

â)   log2 + log ⁡ ( 2 + 2 ) + log ⁡ ( 2 + 2 + 2 ) + log ⁡ ( 2 - 2 + 2 ) = 2 ⁢ log ⁢ 2
0.  Äåßîôå üôé:    log9 · log 4 ⁢ 10 log 4 ⁢ 3 = 2
0.  Íá áðïäåé÷èïýí ïé éóüôçôåò:  á)   log 2 ⁢ 5 · log 25 ⁢ 8 = 3 2               â)   log 5 ⁢ 7 · log 7 ⁢ 5 = 1
0.  Áí   log 4 ⁢ 3 = α    êáé    log 4 ⁢ 5 = β    íá õðïëïãéóôåß óõíáñôÞóåé ôùí   α , β   ï   log 16 ⁢ 15.
0.  Áí   log 5 ⁢ α + log 5 ⁡ ( log 5 ⁢ α ) = 1    ôüôå äåßîôå üôé:   α = 5.
0.  Áí   log α ⁡ ( 2 ⁢ α 2 - 5 ⁢ α + 6 ) = 2    ôüôå äåßîôå üôé:   α = 2   Þ   α = 3.
0.  Áí   α > 0 , 0 < β ≠ 1   êáé   μ ≠ 0 ,   äåßîôå üôé:   log β μ = ν μ ⁢ log β ⁢ α .
0.  Áí   log α ⁢ β = log β ⁢ γ · log γ ⁢ α ,   äåßîôå üôé:   α = β   Þ   αβ = 1
0.  Áí   1 log 2 ⁢ α + 1 log x ⁢ α + 1 log y ⁢ α = 0 ,    äåßîåôå üôé:    x ⁢ y = 1 2
0.  Áí   log ⁡ ( x 3 ⁢ y 2 ) = α   êáé   log x y = β ,   íá âñåßôå óõíáñôÞóåé ôùí   α , β   ôïõò   x , y .
0.  Íá áðïäåßîåôå üôé:

á)    log β ⁢ α = 1 log α ⁢ β        

â)   1 log 2 ⁢ α + 1 log 3 ⁢ α = 1 log 6 ⁢ α

ã)   1 log 2 ⁢ α + 1 log 3 ⁢ α + … + 1 log ν ⁢ α = 1 log ν ! ⁢ α
0.  Äåßîôå üôé ïé èåôéêïß áñéèìïß   α , β , γ   åßíáé äéáäï÷éêïß üñïé ãåùìåôñéêÞò ðñïüäïõ üôáí êáé ìüíï üôáí ïé áñéèìïß   logα , logβ , logγ   åßíáé äéáäï÷éêïß üñïé áñéèìçôéêÞò ðñïüäïõ.

Converted by Mathematica      April 11, 2013