1º bachillerato ecuaciones logarítmicas, neperianas y exponenciales

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- 1 -

ECUACIONES ECUACIONES LOGARÍTMICAS

2

3

2

14

2

12

2

12262log

2

13log22log2

2log62log3log2log2log2log9log4

1log64log

232

22

12

32

26

22322

=−=−=−−−=−−

−=−−+=−−+ −

9

13513log32log51log3log2log1log27

1log

32

1log 322

33

52232

−=

=−−−=−−−=−+=−+ −−

6

1

32

1

32

12

32

11

log3

log2

11

log3

log2

1log1

log

loglog

log

log1

log

3

2

11

3

−=

−

=

=

+−

=+−

=

+−=

+−=+=

+ −

a

a

a

aa

a

aa

a

aa

22

1

2

125,0

2

125,0log

2

2

1 =→

=

→=

→= yyyy

310100001,0100001,0log 3 −=→=→=→= − yx yy ECUACIONES NEPERIANAS

511

ln 555

−=→=→=→= − yeee

eye

yy

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5

2

5

2lnln

5

2ln5ln2ln25ln2ln25ln

2

12ln4ln

44444 =→

=→

=−=−=−= xxx

( ) ( ) 4.15244.1484454ln 5554ln =+=+=→=−→=→=− − exexeex x

( )( )

64.01

44.0ln3

31.1ln

31.1ln331.1ln3

931.1ln

3

318631.1ln

3

263

2ln5ln3

2ln6ln35ln

3

2ln62lnln5lnln6

2ln

44.044.0ln

33 23

≈=→=→−=→−

=→

→=−→=

−→=

+−→=

+−

→−=+−→=+−−→=+−

−

exeexx

xxxx

xxxxxxxxx

x

ECUACIONES EXPONENCIALES

2445122322 22511 22

±=±=→=→=+→=→= ++ xxxxx

( ) 422555

15

25

15 2

2

2

1

2

12

2

2

1

=→−=

→−=→=→=→= − xxxxxx

→=+−→=→= +−+− 0653313 206565 22

xxxxxx

( ) ( )

=++=

=−==→±=−±=

−−±−−=→=+−

32

15

22

15

2

15

2

24255

1·2

6·1·455065

2

12

2

x

xxxx

65,05,0325,0 6 =→=→= xxx Descomposición factorial de 32:

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( )

3333*

277

1891897

9

189

9

3921

93

213·3·213·33·3321333

3

2132121

=→=→=

==→=→=+−→=−−

→=+− →=+−→=+− −−=−−−−

xy

yyyyyyy

y

yyy

xx

yxxxxxx x

22242

4123122122·2212222

211

=→=→==→=→=+ →=+→=+ =+

x

yyyyxx

yxxxx x

( ) →=++ →=++→=+− = 045042·52042·52 2222 yyyxxxx x

( ) ( )

=++=

=−==→±=−±=

−−±−−=→=+−

=+−

∗

∗

42

35

12

35

2

35

2

16255

1·2

4·1·455045

045

2

12

2

2

y

yyyy

yy

Si y =1 02212 0 =→=→=→ xxx Si y =4 22242 2 =→=→=→ xxx