ALGEBRA PREUNIVERSITARIA

1. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
EJERCICIOS RESUELTOS DE ALGEBRA
PREUNIVERSITARIA
LOGARITMOS Y ECUACIONES LOGARITMICAS
Ing. WIDMAR AGUILAR, Msc
Abril 2021

2. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.

3. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.

4. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.

5. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
55)
Cambiando de base:
= .
√
.
√
=
= =
= 2 → ( )
56)
E = = = =
= − − − − − ( )
28 = + 1
14 %&
= 28 → 14. 14&
= 28
14&
= 2 ; 14 = 2
= ( ∗ )
=
%
---------(1)
De (a) : = →
%
=
%
− − − −( )
(1) en (b):
%
=
&
→ 2 + =

6. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
2a = (1 − )
=
&
)&
→ (*)
57)
& = 2 → =
+* = 3 → = *
&
-
=
+
+. → = * − − − (1)
(1) en E:
= &.( * ) = &.( / /)
= &.( 0 /)
= &.
0
+ &.
/
=
0
& +
/
&
= (
0
) ∗ 2 +
/
∗ (1)
= (
0
) +
/
=
/
→ ( )
58)
= ( 3) 1 2
.
33 ( .5)( 4)

7. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= 5 3 6 72
.
38 ( 5) 5 46
= ( 3)(3 3)( 5)( 4)
= (1)( 3)( 5)( 4)
= ( 3)( )( 2 )
== ( 5)(2 2)
= ( 5). =
= → (*)
59)
9 = : = (: + 9) = ;
9 = ; → 3<
= 9
: = ; → 6<
= :
: + 9 = ; → 12<
= : + 9
12<
= 3<
+ 6<
>
> =
>
> +
>
>
4<
= 2<
+ 1
(2<
) − 2<
− 1 = 0

8. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
2<
=
±√ %
=
±√
A
B
=
>
> = 2<
; 2<
> 0
A
B
=
%√
→ ( )
60)
= 2 2. 2
2
. + 7
= E 2. + 7 = .
)
2+7
= − (1) + 7 =
= 6 → (*)
61)
3F
=
9F
= → (3F
) = → =
= + = &
= &
= → (H)

9. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
62)
- + - =1 → - = 1
- − 1 = 0
- − -* = 0 → log 5
&+
-
6 = 0
log 5
&+
-
6 = log 1 →
&+
-
= 1
= * − − − − − (1)
De: LB = - *B)
+ - *B)
LB = -[ *B)
. *B)
]
LB = -[ * B)
] , reemplazando (1):
LB = -[*. * B)
]
LB = -(* B) )
LB = 29-1 ---------(2)
Como: E =
F2%O %O.% )))))))) % OP
B
L + Q + Q + − − − − − − − − + QB =
De sumatorias se tiene:
L = 1
L = 3
L = 5
------
-------

10. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
LB = 29 − 1
∑ 2LS − 1 = (1 + 3 + 5 + 7 + − − − + (29 − 1
B
ST )
∑ 2LS − 1 = 9
B
ST
E =
F2%O %O.% )))))))) % OP
B
=
B
∑ 2LS − 1
B
ST
E =
F2%O %O.% )))))))) % OP
B
=
B
(9 )
= 1 → ( )
63)
log L = √64 − 125
2
. + 10 ; x> 0
log L = 8 − 5 + 10
log L = log (
/
) + 10
log L = log 5
/
∗ 106 = 16
L = 16
S = 1+6 = 7 → (*)
64)
= ( ( (12 ∗ 3) + 108)
= ( ( 6 ) + 108)

11. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= ( 2 + 108)
= U (2 ∗ 108)V
= ( 216)= ( 6 )
= 3=1 → ( )
65)
= 14
2
WXY. . 9
2
WXY .3
Z
= (4 . 9 . ) Z
= (3 . 2 . ) Z
= (3 . 2 . ) Z
= (3 . 2 ) Z = 36 Z
= 5 Z = 5 Z
= 5 = 25 → ([)
66)
F = √2 →
2F
= √2 → L = ( )
2
√
2L =
√
; 2 = √8 → = 5 6
√/

12. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= ]L
=
5
2
6
√^ ( )
2
√
=
2
√
√/
2
=
√
(1) = → ( )
67)
a >0 ; b> 0, b ≠ 1; = 3 → 2 =
+4 = 2 → = 4 → = 2 . 2
= 2. 2
.
= 2 %
.
= 2
`
= √2 . 2 = 4√2
68)
^
=
&
+
WXY `
WXY ^
=
&
+
→
WXY 2`
WXY ^
=
&
+
WXY 2`
WXY ^
=
&
+
→
/.
=
&
+
/
=
&
+
→ . =
&
+

13. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
&
+
↔ = → (H)
69)
= b
. WXYc`% WXYc2
WXYc
=
. WXYc`% WXYc2
WXYc
=
. WXYc`
WXYc
+
WXYc2
WXYc
= 4 +
WXY2
WXYc
WXY
WXYc
= 4 + 5
WXY2
WXYc
)
WXY
WXYc= 4 + 5
2
WXYc
( ) )
= 4+ 5
2
WXYc
(
2
)
= 4 + 5
2
WXYc
( )
= 4 + 5
= 3 → ( )
70)
= E2( √ 00 2
`
7 + 2
.
1 25 3)
= E2( √ 0 2
`
7 + 2
.
(2 ))

14. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= E2( √ 0 10 + 2
.
4)
= E2( √ ( ) 010 + 2
.
2 )
= E2( √ ( ) + 2
.
2)
= E2( 2 (1/2) + 6)
= E2(2 + 6)
= E (8) = E 2
= - → (H)
71)
= 2 .
.WXY .
= 2 .
WXY ..
= 2 .
WXY c
= 2 .
WXY
= 2 .
2
= 2 .
.(
2
)
= 2 .
.
= 2
.
= √2
= √8 → ( )
72)

15. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
e = .F
ZF
=
WXYf
WXY.
WXYf
WXYZ
=
e =
( . )
=
%
De: 3 = g → g = →
h
=
%h
h
=
%h
h
%
→
%h
h
= e
K =
%h
h
→ (*)
73)
E=
00
F
. log L (1) ; x> 0 , x+1 > 0
= log 100 = 010
= 2 → ( )
74)
E= &P
√&
i A
√
P
=
&
2
i
jP
2
P
%A
=
&
iPj2
i
iPj2
P

16. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
iPj2
P
iPj2
i
=
A
B
→ (*)
75)
F]2 =x → (L) = L
= L)` ---------(1)
= F]
kF√]
`
√]
.
= F]
F
2
`.]
2
2l
]
2
.
= F]
F
2
`
]
2
.
E
2
2l
= F]
F
2
`
]
c
.l
= F]L
2
` − F]
c
.l
= F]L −
0 F]
=
fmF) fm]
0
(1)En E:
= f.f
E
`
F)
ff
E
`
F
E`
0
=
0
n 6
F
2
`
L − 7
F
2
`
L)`o
=
0
p 6. 2
`
FL − 7.
)`
2
`
FLq
=
0
r 30. FL + 7.
0
FLs

17. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
0
r 30 (1) + 7.
0
(1)s
= 1 +
=
76)
= 3 → = 2 − − − −( )
/ 5
F ]`
6 = 6
f m`
/
= 6 → log (
F ]`
) = 6 8
log (
F ]`
) = 8
F ]`
= 8 = 2 /
L = 2 → L (2 ) = 2
L = 2 → L = ±4
Como : = 8
|F|
]
=
/
= → (*)

18. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
77)
= √ 0
. √2.4
`
En base 10:
=
√ .
`
√ 0
. =
Y(2l
)
2
`
Y( 0)
2
.
=
2
`
(2l
)
2
.
(
.ll
)
=
( ) 0)
( 00) )
=
( (/. )) )
( ( 00∗ )) )
=
( /% ) )
( 00% ) )
=
( ( .. )) )
( ( 00∗ )) )
=
( % ) )
( 00% ) )
=
% )
( )% )
=
F% ])
]) F% 0
=
]% F)
]) F% 0
→ (H)
78)
& * = LB
→ 1 + & * = LB
+ 1

19. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
& + & * = LB
+ 1
LB
+ 1 = & *
+ * = B
→ 1 + + * = B
+ 1
+ + + * = B
+ 1
B
+ 1 = + *
- = uB
→ 1 + - = uB
+ 1
-* + & = uB
+ 1
uB
+ 1 = - *
=
B
[ b
v&+-
+
w&+-
+
x&+-
P
=
B
[ k &+- + &+- + &+-*
P
=
B
[ k &+-( *)
P
=
B
y √ 1
P
z = 5B
6 (1)
=
B
→ (H)
79)
L = 8 → L = 2 → L = 3
= 24 → = E22 → = −2
F(L + ) =
= ( ) = (3 − 2)

20. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= 1
= 0 → ( )
80)
L =
√
= (
√
) = ( )
L = 2)
= −
2 = ] → = → = ±
Y > 0 → =
E =x+y
= −
= 0 → (*)
81)
(L + 1) + (L + 2) = 1
[(L + 1)(L + 2)] = 2
(L + 1)(L + 2) = 2
L + 3L + 2 = 2
L + 3L = 0
L(L + 3) = 0

21. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
L = 0 ⋎ L = −3
S = 0 -3
| = −3 → ( )
82)
(
f
+ 2)( L − 2) = 0 ; x> 0 y x≠ 1
( L + 2)( L − 2) = 0
( L + 2)( L − 2) = 0
}
L = −2
L = 2
L = −2 → 3)
= L → L =
L = 2 → 3 = L → L = 9
S = + 9
| =
/
→ (*)
83)
(L − 1) − 18 F) 3 = 3 : x-1> 0, x-1 ≠ 1
L > 1 L ≠ 1
(L − 1) −
/
.(F) )
= 3

22. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
U (L − 1)V − 3 (L − 1) − 18 = 0
(L − 1) =
±√ %
(L − 1) =
±
}
(L − 1) = 6
(L − 1) = −3
(L − 1) = 6 → 3)
= L − 1
L − 1 = → L =
/
(L − 1) = 6 → 3 = L − 1
L − 1 = 729 → L = 730
El mayor valor → 730 → (H)
84)
927 BF
− 93 BF
= 2
~
L)F
927 BFf
− 93 B
= 2
~
L)F
9L 93 − 9L 93 = •E2L)F
2 9L 93 − 2 9L 93 = •E2(LF
))
2.3 9L 93 − 2 9L 93 = •(LF
)
4 9L 93 = ln LF
4 9L 93 − xln x = 0
9L(4 93 − L) = 0

23. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
9L = 0 → [0
= L → L = 1
4 93 − L = 0 → L = 4 93
4 93 > 1
L = 4 93 (€ • : •) → (H)
85)
Z ` F
+
`E2F
= ; x> 0, x≠ 1
` F
+
`E2F
=
2
` F
+
) `F
=
` F
−
`F
=
` % `F
−
`F
= →
% `F
−
`F
=
4 L − (1 + L) = (1 + L) L
4 L − 1 − L = L + ( L)
3 L − 1 = L + ( L)
L -1 = ( L)
2( L) − 7 L +3=0
L =
±√ )
=
±
L = 3 → L = 5 = 125

24. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
L = 1/2 → L = 5
2
L . L = 125. 5
2
= 5 . 5
2
L . L = 5
c
→ (H)
86)
L + E2L +
√
L + L = 10 ; x> 0
L − L + L + L = 10
L + .F
.
= 10
L + .F
= 10
2 L + .F
= 10
4 L + L = 20
5 L = 20 → L = 20
3 0
= L
3 = L
L = 81 → ( )
87)
F5.
`5
f
`
6
+
`5
f
Z `
6
= 0

25. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
F5.
`F) `
+
`F) `
= 0
F5.
`F)
+
`F)
= 0
`f
. F
`F)
+
`F)
= 0
L − 4 + L. ( L − 1) = 0
( L) = 4
L = ±2
Si:
L = 2 → L = 5 = 25
L = −2 → L = 5)
=
CS = { , 25} → (*)
88)
LogU√L + 37V = 3 logU√L + 1V
√L + 37 > 0 √L + 1 > 0 → L > 0
LogU√L + 37V = log (kL + 1)
√L + 37 = (√L + 1)
√L + 37 = √L + 3L + 3√L + 1
3L + 3√L − 36 = 0
… = √L → … = L

26. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
3… + 3… − 36 = 0
… =
) ±√ %
=
) ±
… = −4 , … = 3
… = −4 → √L = −64 → 9 *;:† [
, … = 3 → L = 9 − − − ‡ ;*ˆó9 •[
X= 9 → ( )
89)
UFWXY`fV
= 4 : X >0
log(L `F) = 4 log 5
log 1L
WXYf
WXY`3 = log 5
L
WXYf
WXY` = 5 →
F
. L = 4 5
( L) = 4( 5)
log L = 2 5 → log L = log 5
X = 25 → ([)
90)
(3F
− 1). (3F%
− 3) = 6 : x> 0

27. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
(3F
− 1). (3(3F
− 1)) = 6
(3F
− 1). ( 3 + (3F
− 1)) = 6
(3F
− 1). (1 + (3F
− 1)) = 6
( (3F
− 1)) + (3F
− 1) − 6 = 0
Haciendo: (3F
− 1) = ;
; + ; − 6 = 0
(; + 3)(; − 2) = 6
Š
; = −3
; = 2
; = −3 → (3F
− 1) = −3 → 3)
= 3F
− 1
3F
− 1 = → 3F
=
/
Tomando logaritmo de base 3:
3F
= 5
/
6 → L 3 = 5
/
6
L = 5
/
6
; = 2 → (3F
− 1) = 2 → 3 = 3F
− 1
3F
− 1 = 9 → 3F
= 10
3F
= (10) → L 3 = (10)
L = (10)
L + L = 5
/
6 + (10)
L + L = 5
/
. 106
L + L = 5
/0
6 = 280 − 3
L + L = 280 − 3 → (*)

28. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
91)
√ F) (2L − 3) = .3 + 2)
. 2.
(√ F) ) (2L − 3) = 4/3 + 2)
2
.
F) (2L − 3) = −
F) (2L − 3) = 1
F) (2L − 3) = F) (2L − 1)
(2L − 3) = 2L − 1
4L − 12L + 9 = 2L − 1
4L − 14L + 10 = 0
L =
±√ ) 0
/
=
±
/
‹
L =
L = 1
L = 1 → ‡[ H[ •ˆ…: √ F) (2L − 3) [‡ < 0
Y no existe
L = → ( )

29. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
92)
6 + 5 L = ( f )
; x> 0 , x ≠ 1
6 + 5 L = (
f
)
6 + 5 L = ( L)
; = L
; − 5; − 6 = 0
; =
±√ %
=
±
}
; = 6
; = −1
Luego:
L = 6 → L = 2 = 64
L = −1 → L = 2)
=
L . L = 64.
L . L = 32 → (H)
93)
log (√L + 2L + 3 = logU√L − 1
.
V)

30. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
L + 2L − 3 > 0 → L < −3 • L > 1
log(L + 2L + 3)
2
= log(L − 1)
.
.
2
.
(L + 2L + 3)
2
= (L − 1)
2
L + 2L + 3 = L − 1
(L − L )- (2x-2) = 0
L (L − 1) − 2(L − 1) = 0
(L − 1)(L − 2) = 0
}
L = 1 ó
L = 2 → L = ± √2
− √2 … … 9 *;:† [
1 -------- vuelve infinito a unos de los logaritmos,
No cumple
L = √2 → ( )
94)
√F) √L + L = 2
√F)
√L + L = 2
F) L + L = 2
(L − 7) = L + L
L − 14L + 49 = L + L
15L = 49 → L =

31. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
L = → ‡[ H[ •ˆ…: [‡ 9[ …ˆ€
Y por tanto no existe
→ 0 ‡ ;*ˆ 9[‡ → ([)
95)
|L| + L + 4 = 4 U√ FV
log L + L + 4 = √2L
log L + L + 4 = (√2L)
log L + L + 4 = 2L
log L = L − 4 ; X > 0
Graficando las ecuaciones: se ve se cortan en dos puntos
Soluciones = 2 → (*)
96)

32. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
(5
2
f + 125) = 30 +
F
(5
2
f + 125) − 30 =
F
7
2
f%
0
8 =
F
7
2
f%
0
8 = 5
2
f
2
f%
0
= 5
2
f
2
f
0
+
0
= 5
2
f
7
2
f
0
+
0
8 = 55
2
f6
(
2
f)
00
+
0.
2
f
00
+
00
= 5
2
f
55
2
f6 − 650. 5
2
f + 15625 = 0
5
2
f =
0±√ 0 ) 00
=
0± 00
5
2
f2 = 625 → 5
2
f2 = 5 →
F2
= 4
L =
5
2
f = 25 → 5
2
f = 5 →
F
= 2
L =
L + L = +
L + L = → ( )

33. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
97)
5 U ( L)V6 = 5 : x >0
U ( L)V = 5
U ( L)V = 4
( L) = 4
( L) = 1024
( L) = 3 0
L = 3 0
X = 2
2l
→ ([)
98)
(2L + 15L + 26) = 4
2L + 15L + 26 = 4
2L + 15L − 38 = 0
L =
) ±√ % 0
=
) ±
L = 2 ; L = −
L +L = 2 −

34. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
L +L = − → ( )
99)
& %+2( + − 4) = 2
( + ) = 2 + 2 − 8
+ 2 + = 2 + 2 − 8
− 2 + ( − 8) = 0
( ) − 2 + ( − 8) = 0
=
+±k + ) (+ )/)
=
+±√
= ± 2√2
− = ±2√2 → | − | = 2√2
• = k| − |
.
• = b(2√2)
.
• = ((2√2) )
2
.
• = ((25) √ )
2
. = ((25)
./
)
2
.
• = ((25)
.
)
2
. = (5 )
2
• = 5 → ([)

35. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
100)
log(L + 2 L) = log(8L − 6 − 3)
L + 2 L = 8L − 6 − 3
L + L2( − 4) + 3(2 + 1) = 0
L =
/) & ± k (&) ) ) ( &% )
Si el discriminante: ∆ = 0 → ‡ ;*ˆó9 •[ ú9ˆ*
4( − 4) − 12(2 + 1) = 0
4( − 8 + 16) − 24 − 12 = 0
− 8 + 16 − 6 − 3 = 0
− 14 + 13 = 0
=
±√ )
=
±
= 13 → L = −9 → log(8L − 6 − 3) → ∞
= 1 → L = 3
= 1 → ( )
101)
1+2 |L| − log(L + 2) = 0
log(L + 2) − 2 |L| = 1

36. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
log
F%
F
= log 10
F%
F
= 10
10L − L − 2 = 0
L =
±√ %/
0
=
±
0
‹
L = 1/5
L = −
0
L + L = −
0
L + L =
0
→ ([)
102)
( L) − /L = 8 ; x>0
(
F
) −
F.
. = 8
(
F
) −
F.
= 8
( F)
− L = 8
( F)
− L = 8
( L) − 4 L = 32
… = L
… − 4… − 32 = 0

37. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
… =
±√ % /
=
±
… = 8 → L = 8 → L = 2/
… = −4 → L = −4 → L = 2)
L . L = 2/
. 2)
L . L = 2 = 16 → (H)
103)
5
F
6 + ( 25L) = L + 7
( L − 5 ) + ( L + 5 ) = L + 7
( L − 3) + ( L + 2) = 6 L + 7
( L) − 6 L + 9 + ( L) + 4 L + 4 = 6 L + 7
2( L) − 8 L + 6 = 0
( L) − 4 L + 3 = 0
L =
±√ )
=
±
L = 3 → L = 5 = 125
L = 1 → L = 5 = 5
L + L = 125 + 5
L + L = 130 → ( )

38. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
104)
(L − 1) + (L − 3) = 2
[(L − 1)(L − 3)] = 2
(L − 1)(L − 3) = 2
L − 4L + 3 = 8
L − 4L − 5 = 8
L =
±√ % 0
=
±
L = 5 ; L = −1
L = −1 → 9 *;:† [
L = 5 → (*)
105)
F(5 − L) +
fj F
= F6
F(5 − L) + F(L + 2) = F6
F[(5 − L)(x+2)] = F6
(5 − L)(L + 2) = 6
-L − 3L − 10 = 6
L − 3L − 4 = 0

39. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
L =
±√ %
=
±
}
L = 4
L = −1
Como la base x > 0 y x≠ 1 → L = −1 9 *;:† [
L = 4
= L
= . =
= → ( )
106)
&64. F = 6 +*. -L. F
Por regla de la cadena:
&64. F = 6 +
&64. F = 6
&
.
&
F
=6
F
= 6
log 64 = 6. L
log 64 = log L
2 = L
L = 2 → 2 =
= 1 → ( )

40. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
107)
25 f = (L − 5L + 15) f ; x >0 , x ≠ 1
3 f = (L − 5L + 15) f
3 f = (L − 5L + 15) f
3 f = (L − 5L + 15) f
(3 ) f = (L − 5L + 15) f
9 f = (L − 5L + 15) f
L − 5L + 15 = 9
L − 5L + 6 = 0
L =
±√ )
=
±
}
L = 3
L = 2
=
F2.F
%F2%F
=
( )
% %
=
E = 1 → ( )
108)
9L. L + 9[. = 9L •

41. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
9L. L + = 9L •
( 9L) − 2[ 9L + = 0
3( 9L) − 6[ 9L + 4 = 0
9L =
•±√ • ) /
=
•± √ • )
9L = e ± √9[ − 12
L = [•%
2
.
√ • )
L = [•)
2
.
√ • )
L . L = [•%
2
.
√ • )
. [•)
2
.
√ • )
L . L = [•%
2
.
√ • ) %•)
2
.
√ • )
L . L = [ •
→ (H)
109)
.F
+
c 0F
+
“E2 f
2l
= 0
.F
+
.. 0F
+
.E f
2l
= 0
.F
+ 2
. . 0F
+ E2
.
f
2l
= 0
.F
+
. 0F
−
.
f
2l
= 0
.F
+
. 0% .F
−
.F) . 0
= 0

42. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
( .F) )( . 0) % .F( .F) . 0)) .F( .F% 0)
.F.(( .F) )( .F) )
= 0
2 ( L) − 5 L. 10 − ( 10) = 0
L = . 0±k ( . 0) %/( . 0)
L = . 0±k ( . 0)
L = . 0± √ . 0
L = 10 +
√
10
L = 10
`
+ 10
√..
L = (10
`
. 10
√..
)
L = 10
`
%√..
L = 10
`
− 10
√..
L = (10
`
. 10
√..
)
L = 10
`
)√..
L . L = 10
`
%√..
10
`
)√..
L . L = 10
`
%
`
= 10
`
L . L = √10 = 100√10
L . L = 100√10 → (*)

43. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
110)
L F
=
0Z
F
Tomando logaritmos de la expresión:
log (L F
) =
0Z
F
L. log L = log 10 − log L
( L) + log L = 6
( L) + log L − 6 = 0
( L + 3)( L − 2) = 0
}
log L = −3
log L = 2
log L = −3 → L = 10)
log L = 2 → L = 10
L . L = 10)
. 10 = 10)
L . L =
0
→ ([)