Ecuaciones Exponenciales y Logaritmicas

�� 5: �� 0

=1

1

=

1) a 2) a 3) a 4) a 5) a

−n

m

6) a : a 7) ( a

a

1

=

a m

m

n

=

8) a

n

n

a

n

⋅a =

)

n

=

=

n

⋅b =

a

m+ n

m− n

a m⋅n

( a ⋅ b)

a 9) n =   b b a

m

a

n

m

n

n

n

n

� �� .

a x

=b

�� : �: 1) a > 0

a ≠1 a ⇒m=n

2) a

m

=

3) a

m

=b

n

m

⇒a=b

�� :

�� , �� . �� . x +1

5 −8

=5

−4 x +13

⇒

−8 x + 1 = −4 x + 13

⇒

−12 =

4 x ⇒ x = −3

�� , �� .

2 ⋅ 32 x −5

=

54 ⇒ 2 ⋅ 32 x −5

=

2 ⋅ 33 ⇒ 2 x − 5 = 3 ⇒ 2 x = 8 ⇒ x = 4

�� , �� , �� , �� . �� . x

22 x

�� 2

= t

�� :

t

2

x

− 5⋅2 + 4 = 0 − 5 ⋅ t + 4 =

0

⇒

t = 4 t = 1

⇒

2 x

=

2 x

=1

4

⇒

x = 2 x = 0

1

�� 1. �� . x

�. 5

2

−5 x + 6

3 x

=1

1− x2

�. 3

1

=

= 0 '5

x + 2

�. 7

4 − x2

�. 3

= 5764801

�.

27

3 x+ 2

�. 2

1

1

=

4 x −

= 186

2 x+ 2 x2 − 2

�. 3

9

=

1 3

2. �� . �. 4

2 x

x

�. 3

− 2 ⋅4

+3

x+2

x +1

x

+ 16 = 0

�. 2

x

�. 2

= 30

2x+1

+

= 12

1− x

=3

2 x − 5

=

+2

x +1

�. 5

x

+5 +5

x −1

=

31 5

3. �� . �. 5

2 x

x

�. 2

3

⋅2

x +1

�. 5 ⋅ 7 x

�. 7 ⋅ 3 �. 4

2 x −1

x

�. 2

0'2

=

�. 3

x

=

2

+1

4 x− 6

�. 3

x

�. 0 '5

9

− x

�. 2

=8

x−1

x

+3

x+ 2

x + 2

= 823543

�. 7

+3

x− 2

�.

= 21

x

�. 7

= 39

�. 5

= 32

�. 3

= 390625

2

=1

+1

= 3125

25 x + 2

x

5 x − 2

3+ x

5 x

x

−5⋅2 + 4 =0

−3

�. 3

0 ' 25

�. 10

= 16

x

�. 3

= 35

= 567 =

2 x

2187

1

=

49

−3

−x

=

778 27

4. �� .

 2 x + 3 y �.  x 1 y 2 − 3 +

7 x y = 493 �.  x y  7 = 49 +

=7 −1

⋅

=7

 2 x + 3 y = 7 �.  x 1 y 1 2 − 3 = −1

 2 x y = 128 �.  x y  2 + 2 = 24

 2 x + 3 y 1 = 5 �.  x 1 y 1 2 + 8 ⋅ 3 = 7

 2 x y = 4 �.  x y  2 − 2 = 768

 x − 4 y = 5 �.  x 6 y 2 ⋅ 2 = 16

5 x y = 1253 �.  x y  5 = 125

+

+

−

−

+

−

+

 x 433  2 = y �.  4  x − y = 0  33  4 x  2 = y �.  4 4 x − 4 1 ⋅ y = 4 0  −

+

−

−

� �� , �� , x

2

=

7

�� , �� x �� , ��

4 ≤ 7 ≤ 8 ⇒ 22

x

≤2 ≤2

3

⇒ 2 ≤ x ≤ 3

�� x . �� x

�� . 2

=

7

��: �� a �� , � �� x �� a �� . �� : x log a b = x ⇔ a

=

b

2

a �� , �� , a > 0, a ≠ 1 . � b �� , �� , b > 0 . ��: ��

�� 3 �� 81; �� :

log 3 81 = x ⇒ 3 x

= 81 ⇒

x = 4 . �� 3 �� 81 �� .

��

�� 10 �� 1000�

log10 1000 = x ⇔ 10 x

= 1000 ⇔

x = 3 . �� 10 �� 1000 �� .

��: �� 10 �� 1)�� .

log a ( b ⋅ c ) = log a b + log a c 2)�� .

b log a   = log a b − log a c c 3)�� .

log a b n

=

n ⋅ log a b

4)�� .

log a b =

log c b logc a

5)�� : �) log a 1 = 0 �� a

0

=1

1

=

�) log a a = 1 �� a

�� a .

a �� a .

�� : �� x = log a b , y

=

x log a c � z = log a ( b ⋅ c ) , �� a z

1)�� : a

= b⋅c =

a

x

⋅a

y

=

a

x+ y

=b,

a

y

=

z

c �a

= b⋅c

⇒ z = x + y ⇒

log a ( b ⋅ c ) = z = x + y = log a b + log a c ⇒ log a ( b ⋅ c ) = log a b + log a c b b b  ⋅ c  = log a + log a c �� log a c c c 

2) log a b = log a  3) log a b

n

= log a

x

x =

=

b − log a c

( b ⋅ b ⋅ ... ⋅ b ) = log a b + log a b + ... + log a b = n ⋅log a b

4)�� x = log a b �� a

log c a x

= log a

=b

�� .

log c b �� 3) �� x ⋅ log c a = log c b � �� x ��

logc b logc a

�� log a b =

log c b logc a

3

�� 1) ��, �� , �� . �) log 2 8

�) log 0 '0001

�) log 3 81

�) ln e

33

�) log 1000

�) ln e

−4

�) log 4 16 �) log 4 0' 25 �) log 4 0' 0625

2)�� log 2 = 0'3010 ; log3 = 0'4771 � log7

= 0'8451 ,

��

10 �� . �� , �� log3'5 � �� log1'5 � 3)�� log a b ⋅ log b a . 4)�� . �) log3 x =

2

�) log 5

6

625

=

�) log x 3 = 2

x

�) log x 256 = −8

3

5)�� x

�) 2 '3

�) 3 + 2 ⋅ log x = 5

= 18

�) log x = log 9 − log 4

�)

�) ln x = 3 ⋅ ln 5

2 x

=

3

�)

1 3

⋅ log 2 x = −3

7 '5

6)�� ) log x = log15 − log 2 + log 3

�) log 2 x = −3'5 �) log 3 x

3

=

�) log ( x − 3 ) + log x = log ( 4 x )

13 5

�) 2 ⋅ log x − log ( x − 16 ) = 2

�) ln ( x − 3) = 2

�) 4 ⋅ log 2 ( x

�) log x 169 = 2

2

)

+ 1 = log 2 81

7)�� ) log 3 x − 3log 3 x + 4 log 3 x �) log 2 x − log 2 x �) 2 ln x − 4 ln

2

= log 4

( ) x

2

�) log x

=2

�)

2

+ 7 log

1 =7  x 

�) log ( 5 x + 1) − log 2 =

+ ln 

log( 5 − x )

1 2

log ( x + 4 )

�) log ( 3 x − 1 ) − log ( 2 x + 3 ) = 1 − log 25 �) 2 + log ( x − 2 ) = 3 1 − log ( x − 2 )

x −9 = 0

log 2 + log (11 − x 2 )

+ log100 = 3

�) 2 log x − log ( x − 16 ) = 2

x

�) log 2 + log x = 1 �) ( log x )

2

=

2

�)

log ( 35 − x3 ) log( 5 − x )

=3

�)

log 2 + log ( x 2 log

(

5 − x

−5

)

)

=

2

8)��

log x + log y = 30

�) 



x + y = 60

 3log x − log y = 1

�) 

log x + 2 log y = 5

log x − log y = 1

�) 

2 2  x − y

=

4


�) 



x + y = 29


�) 



x − y = 6

log ( x + y ) − log ( x − y ) = log 5  �)  2 x =2   2 y 4

Ecuaciones Exponenciales y Logaritmicas

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