martes, 14 de enero de 2014

LOGARITMOS

EJERCICIOS DE LOGARITMOS

I )  Calcular :
 1 ) log 2 8   =                                                                                                            R :  3
 2 ) log 3 9   =                                                                                                            R :  2
 3 ) log 4 2   =                                                                                                            R :  0,5
 4 ) log 27 3   =                                                                                                          R :  1 / 3
 5 ) log 5 0,2   =                                                                                                         R :  - 1
 6 ) log 2 0,25   =                                                                                                       R :  - 2
 7 ) log 0,5 16   =                                                                                                        R :  - 4
 8 ) log 0,1 100   =                                                                                                      R :  - 2
 9 ) log 3 27   +   log 3 1   =                                                                                       R :  3
10 ) log 5 25   -   log 5 5   =                                                                                      R :  1
11 ) log 4 64   +   log 8 64   =                                                                                    R :  5
12 ) log 0,1   -   log 0,01   =                                                                                     R :  1
13 ) log 5   +   log 20   =                                                                                           R :  2 
14 ) log 2   -   log 0,2   =                                                                                          R :  1
15 ) log 32 / log 2   =                                                                                                R :  5
16 ) log 3 / log 81   =                                                                                                R :  0,25
17 ) log 2 3  ´  log 3 4   =                                                                                          R :  2     
18 ) log 9 25  ¸  log 3 5   =                                                                                        R :  1

 II )  Determinar el valor de   x :
 1 ) log 3 81   =   x                                                                                                     R :  4
 2 ) log 5 0,2   =   x                                                                                                    R :  - 1
 3 ) log 4 64   =   ( 2 x  -  1 ) / 3                                                                                R :  5
 4 ) log 2 16   =   x 3 / 2                                                                                             R :  2
 5 ) log 2 x   =   - 3                                                                                                   R :  1 / 8
 6 ) log 7 x   =   3                                                                                                       R :  343
 7 ) log 6 [ 4 ( x  -  1 ) ]   =   2                                                                                  R :  10
 8 ) log 8 [ 2 ( x 3  +  5 ) ]   =   2                                                                                R :  3
 9 ) log x 125   =   3                                                                                                   R :  5
10 ) log x 25   =   - 2                                                                                                R :  1 / 5
11 ) log 2 x  +  3 81   =   2                                                                                            R :  3
12 ) x  +  2   =   10 log 5                                                                                             R :  3
13 ) x   =   10 4 log 2                                                                                                    R :  16
14 ) x   =   log 8 / log 2                                                                                             R :  3
15 ) x   =   log 625 / log 125                                                                                     R :  4 / 3
16 ) log ( x  +  1 ) / log ( x  -  1 )   =   2                                                                   R :  3
17 ) log ( x  -  7 ) / log ( x  -  1 )   =   0,5                                                                R :  10
III.  Determina el valor de x:
a) logx(1/25) = -2
b) log0,52 = x
c) logxo,25 = 0,5
d) log2(1/4) = x
IV. Expresa como un solo logaritmo:
a) -log a - log b
b) log a + 2 - 2log b
V. Dados log 2 = 0,3 y log 3 = 0,47, calcula:
a) log 81
b) log 15
c) log 0,75


VI. Transforma a la forma exponencial y calcula x. 
a) log2x = 4
b) loxx81 = 4
c) logx(1/8) = 3
d) log1/2x = -3
e) log264 = x
f) log4x = 3/2


VII. Desarrolla, aplicando las propiedades de los logaritmos

a) log (3ab)
b) log (5a/2)
c) log (4a2/3)
d) log (a3b5)
e) log (2/ab)




VIII) Determina x utilizando la definición de logaritmos

a) log2x = 4 b) log5x = 0 c) log3/4x = 2
d) log1/2x = -3 e) loxx81 = 4 f) logx16 = -4
g) logx(1/8) = 3 h) log264 = x i) log3(1/81) = x
j) log4x = 3/2 k) logx4 = -2/5 l) log1/64x = 5/6

IX. Desarrolla aplicando las propiedades de los logaritmos:

a) log (3ab)           b) log (5a/2)                          c) log (4a2/3)
d) log (a3b5)             e) log (2/ab )

X. Reduce las expresiones siguientes a un solo logaritmo:

a) log a + log b b) log x - log y
c) 1/2 log x + 1/2 log y d) log a - log b - log c
e) log a + log b - log c - log d f) log x - 2 log y + log z
g) 2/5 log a + 3/5 log b h) log a + 1/2 log b - 4 log c
i) 1/2 log a - 2/3 log b + 3/4 log c j) log (x + y) - log 3
 k) 1/3(log a - 3log b) + 1/4(log c - 3log d) l) -log a - log b

XI. Sabiendo que log 2 = 0,3; log 3 = 0,47; log 5 = 0,69  y  log 7 = 0,84; calcula, sólo utilizando estos valores, los siguientes logaritmos:

a) log 4 b) log 12 c) log 81 d) log 42  
 g) log (5/7) h) log 3,5 i) 2log 250 j) (log 18)·(log 16)

XII.Resuelve las siguientes ecuaciones:

1) log 4x = 3log 2 + 4log 3
2) log (2x-4) = 2
3) 4log (3 - 2x) = -1
4) log (x + 1) + log x = log (x + 9)
5) log (x + 3) = log 2 - log (x + 2)
6) log (x2 + 15) = log (x + 3) + log x
7) 2log (x + 5) = log (x + 7)


8) 52x-3 = 22-4x


9) log (x - a) - log (x + a) = log x - log (x -a)

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