Ejercicios resueltos de logaritmos PDF

Title	Ejercicios resueltos de logaritmos
Author	Jaime Angel Ortiz Diaz
Course	Matemática Financiera
Institution	Universidad Complutense de Madrid
Pages	8
File Size	227.3 KB
File Type	PDF
Total Downloads	106
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    LOGARITMOS A. Introducción Teoría     !

B. Ejercicios resueltos "#$% "&!'#$% "() " &!' "*+

A. INTRODUCCIÓN TEÓRICA

              !   " # # "$$   # " % $ " #   # " %    





,-./0.,+    #& &        $   '( )         *+             ,  ,    %- .

  ,/

..

  , 

...

 

, /

  ,   /     , 

  ,   / 

.0

   1 

0

  

0.

  

 



B. EJERCICOS RESUELTOS B.1. Dado un logaritmo, halla su valor: 

 # " % = # # " = " ⋅ # # = " ⋅  = " 



      # # = # # # = ⋅ # # = ⋅  =  # # #



  

−

        # =   # # = ⋅   # = ⋅     = ⋅(− )     = −      # # # # # # # # # #

#



 % 2

  3 =   * =   ( * 2

2

*

*

%

)

*

−

% %   =   * = ⋅   * = ⋅     =    2 2 * * * *

  % % % = (−) ⋅     = − ⋅  = −    2 2 2 * * 



% 2



,-./0.,+    #

 1 ( 2  1 11) = #  1 (2  1 11) = #  1 ( 2 1 1# ) = 



= #  1 ( 2 ⋅ #  1 1) = #  1 1 = # 





#

*# = 

  1   # =    # #  = 1 ⋅    # #  = 1      ##  ##  2



##

2

 4 * * ⋅ #5 = 





*⋅ * #

* ⋅( *

 * 2

)

= 

 +

*

*+  2

*#

3# ⋅

 * 2 * =  * * =  *  * #  =  *# *#    3 2

 *  " "  #  =  * *  =   2 2 *#  B.2. Dada una expresión logarítmica, hallar su valor.   −  # # +  # 3 +  # =  # # 2 +  # # * +  # # # =  %   " =  # # + *  # # − #  # # = + * − # =  2 2 2 2



    +   +   * +  *  +  = = 1 +  + *   +   * +  −  =  

 

 $ * ≈ 1$%55  4   *  31 +  1$ 1* +  2 =  (* % ⋅ 1 )+  +  (*− # )2 =  4 11 # # = %  * +  1 +  * −  1# −  * = %  * + +  *− #−  *= 2 2 #* =  * −  = #$ 4%# −  = $ 4%#  2



 31 +  1$ 1* +  2



 2 1$ 1% +  *

  1$ #2 $ " $# ≈ 1$*1  +  3 2 1$ #2 $ "  2 1$ 1% +  * +  = 3 2 



 # *  % #   2   #  #  # 2   +  1 =  =   +   11    *  #   11   2       





,-./0.,+    2#   %    #    #      +    1  =  =    +   11   *     2 11  *  #  #  2         # %       2 #  = (  ## −  1# ) +   −  #*  +   −  2  =   2 *   11   #   1       1 1  = (#  # − #  1) +  #  − #  1 − *  # +  %  # −  1−   =  #  2 *  # #     = (#  # − # ) + # ( − # ) − # − *  #  +  %  # −  − ( −  # ) =  2 * # # # # # #    5 5 =  # − + −  # − − # + #  # − − −  # = − +  # = 2 2 * * * # # # 2 *1 5 5 = − +  # = −$ **  2 *1 #

 *



2

    +   

  2    = +    ⋅  



"

" #  " 2    = +    =  *  

"  % = + =  2 " *1  



 −

*

 +   +  + − 

+ =

−

   =  − (  − ) * +     +  + (  + ) # =  

       =  − ( − )−     +  + ( + ) = −  + = −    # * * # "  



  ( *  ⋅ * ) −

(

   =   * ⋅ *  −   

2

(

#

2

) + 

#

−3

) − * =   

⋅





 



*

− ( ) = 





1 *

 −  

  

−3 2

 1 3 #4 − * = * + 2 − * = 2 

,-./0.,+    −

   −   − ( − ) # +     +     −  =    +  +  + ( + ) #

 − 

=

  −  − ( − ) −     −  −    # = = # = −*    + ( + ) # #     * 2  # +  # # % +  # #− * ( ) # 3= =   # % # −  # # *  #  # % − *  # # * * # # 2 + % # # − * # # 2 + % − * 3 = = =  %  # # − *  # # % − * 2

 # 2 3 +  # " +  # 

 

#  #2  # −  # #2 − 2 3 =  # %1 −  # 1  # # +  # %  # 3 +  #



=

# # * + (# #− # 2# )

( # 2 +  # # * )−(  # 2 +  # #)

−

# 2−  − ( # 2# − # #* )  # # +  # # #

* + ( − #  # 2) −  # 2 −( # # 2 − *) − = + # ( # 2 + * )− ( # 2 + ) % − #  # 2 − *  # 2 + * = − = # −  # 2 −  +  # 2 =   # * =





 # = %  5$ # * ⋅ 1$ 11"− #        * = #   #2 ⋅ *$ # %    2 = −*

 5$ # * ⋅ 1$ 11"− #     =  (5$ # * ⋅ 1$ 11"− # ) −  ( #2 ⋅ *$ # % ) =   #2 ⋅ *$ # %  # # ⋅* # * =    +   2 

−#   *    ( 2# ) +   −    ## ⋅ 2*  

 % %   #   =   2   

= *[ #  # + #  * −  2] − # [  * − #  # − *  2]− 



*

=

,-./0.,+   −  #  2 + %( %  # −  2) =    = * # ⋅ % + # ⋅ # −(− *)  − # # − #⋅ % − * (− *) − { # (− *) + % %⋅ %− (− *) } = 

= *[ 3 + % + *]− #[ # − 3 + 4] − {− " + %[ " + *]} = − *   B.3 Hallar el término desconocido.  

  #2 = * ⇒  * = #2 ⇒  * = 2 * ⇒  = 2 

 

   #%* = 2 ⇒  2 = #%* ⇒  2 = * 2 ⇒  = * 





  "#2 #2 =  ⇒ "#2 = #2 ⇒ 2%  = 2# ⇒ % = # ⇒  =  #



 *# 1$ #2 =  ⇒ *# = 1$ #2 ⇒ #2 =



 2   2  2   # = ⇒  = # ⇒  2  = #2 ⇒ 2 = #2 ⇒  = *#  2  

  # ⇒ #2 = #−# ⇒ 2 = −# ⇒  = −  % 2



B.4. Desarrollar expresiones logarítmicas: ⋅  =  ⋅ −  )=  +    −  )  )



  



  #       = #   = # (    −   )     



  

⋅  =  ⋅ −  )=  +    −  )  ) *





  =    *  −  ) =   * +    −   )# =  )  = *    +    −  )  #  

  



1

,-./0.,+    B.5. Escribir como un solo logaritmo: #



     ( ) − #    =  ()−         



#  ( − )− ( # −

     *     =   #  =          #    

 #

# )= (  − ) −  ( + ) (− ) =

#

#

=  (  − ) −  (  + )(  − ) = 

 − ( − ) =    + ( + ) ( − )

   

 



%  #

= #

 − −  −  #     #

( − ) 

%

#

 − − #   

 %  =  #   −    #

  

  ( − %  = #    −  ( #  



% # %   −     −   =  #         # )    # ( − )#    =  =  #  % # #  −   ( ) )    

  #  #  =  # (−# )    



 #  2 () − 2 ( ) +(  + #) 2 ( 5) =  2  # −  2 * #  # ⋅ 5+# # ⋅ 5 +# = 2  + 2 5 +# = 2 = 2 *   *

 *

+  2 5 +# = 

*





 /                +   +    −    =   ⋅ ⋅ −    =             /                    =   −   =    =    =                         



2

,-./0.,+  

     #   # ( )−  #  #  +  #  =     #  #        #   #   # %     =   *    =       = #   + #   #  #       #    #    #   #   



    TEMAS RELACIONADOS Ecuaciones logarítmicas. Ecuaciones exponenciales.    



...