Aula de logaritmos

Função Logarítmica f(x) = logax, a > 0, a 1 e x > 0

y

x

y

x 1 1

a > 1 Estritamente

crescente

0 < a < 1 Estritamente decrescente

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•logax1= logax2 x1 = x2 > 0 • Se a > 1 : Função estritamente crescente

• logax1 > logax2 x1 > x2 > 0 •Se 0 < a < 1 : Função estritamente decrescente

• logax1 > logax2 0 < x1 < x2



logaN = a = N log10100 = 2, pois 102 = 100 log1/381 = -4, pois (1/3)-4 = 81 log71 = 0, pois 70 = 1 antilog23 = 23 = 8 antilog24 = 24 = 16 colog23 = -log23

y

x 1



logaN = a = N loga1 = 0 logaa = 1 loga a = •log x = log10 x •ln x = loge x {e = 2,71828...}

y

x 1

NaNa

log

Pequenas mudanças no valor de x causam grandes mudanças no

valor de f(x)

Importante !!!


Função Logarítmica Propriedades

loga(M.N) = logaM + logaN loga(M/N) = logaM – logaN loga(Nm) = m.logaN

Nm

NN am

am

a log1

loglog

1

a

NN

b

ba

log

loglog


NaNa

log


loga(M.N) = logaM + logaN log210 = log2(2.5) = log22 + log25 = 1 + log25 loga(M/N) = logaM – logaN log3(1/3) = log31 – log33 = 0 – 1 = -1 loga(Nm) = m.logaN log381 = log334 = 4.log33 = 4



Nm

NN am

am

a log1

loglog

1

3log2

13log

4

23log

4

19log

4

19log9log 22

222

4

1

24

2

2

5log

2log.2

5log

2log

5log

4log

5log5log 2

2

2

22

2

2

24

a

NN

b

ba

log

loglog


Função Logarítmica Exercícios


Simplifique as expressões:



1)2(log)3(log 22 xx

)2(loglog)1(log2 333 xxx

S = {4}

S = {}

1logloglog 842 xxx 7 64S

72 x 7log 2S

0222 xx )12(log 2 S

xxx

93log

9;3

1S





1)(log 2

2 xx 2;1S

8.

4loglog 42

yx

yx

4

1;32S

82

1loglog 22

yx

yx 1;2S

06loglog 2 xx

100;1000

1S

03log)][log( 22 xx



)3(log)31(log 2

117117 xx

1)3ln(2 x

1)3(log)4(log 66 xx

)3(log1)21(log 77 xx

3)6(log)3(log 22 xx

xx 24 log2)1(log21



)93(log)42(log 22 xx 5| xxS

3)1(log 2

2 x 33| xouxxS

12log)3log()2log( xx 1| xxS

)5(log)42(log2

12

1 xx 12| xxS

1)(loglog 33

1 x 3 31| xxS



0ln. xxx exxS |

1,7)10

log(6,53

x

90)10

log(.1012

x

4,5log3,2 x

0ln12

x

x

22 )(ln)ln( xx

1,46,2 1010| xxS

310| xxS

3,24,5 1010| xxS

exxS |

210| exouxxS

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Aula de logaritmos