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Matematika X – Semester 1 SMAN 1 Bone-Bone 19

Bab I Bentuk Pangkat, Akar, dan Logaritma

Standar Kompetensi : 1. Memecahkan masalah yang berkaitan dengan bentuk pangkat, akar, dan Logaritma

Kompetensi Dasar: 1. Menggunakan aturan pangkat, akar, dan logaritma 2. Melakukan manipulasi aljabar dalam perhitungan yang melibatkan pangkat, akar, dan

logaritma

Alokasi Waktu: 26 jam pelajaran (13 x pertemuan)

Indikator Pencapaian Hasil Belajar 1. Siswa dapat mengubah bentuk pangkat negatif ke pangkat positf, dan sebaliknya. 2. Siswa dapat mengubah bentuk akar ke bentuk pangkat,dan sebaliknya. 3. Siswa dapat mengubah bentuk pangkat ke bentuk logaritma, dan sebaliknya. 4. Siswa dapat melakukan operasi aljabar pada bentuk pangkat, akar, dan logaritma. 5. Siswa dapat menyederhanakan bentuk aljabar yang memuat pangkat rasional dan

logaritma. 6. Siswa dapat merasionalkan bentuk akar. Pertemuan ke-10 s.d. 13

Rangkuman Materi

C. Logaritma

Logaritma adalah operasi invers dari perpangkatan. Jika a x = b, maka ditulis dalam bentuk logaritma a log b = x. Dibaca logaritma b dengan basis a adalah x. a disebut bilangan pokok/basis untuk a > 0 dan a ¹ 1, x disebut numerik x > 0 Sifat-sifat logaritma

1. a bba

=log

2. a log (b x c)= a log b + a log c

3. a log ÷øö

çèæ

cb

= a log b - a log c

4. a log b n = n a log b

5. a log b = ab

c

c

loglog

6. a log b x b log c = a log c

7. na log b m =

nm

. a log b

8. a log 1 = 0

9. a log a n = n

Matematika X – Semester 1 SMAN 1 Bone-Bone 20

Catatan : Logaritma dengan bilangan pokok 10, yakni 10 log x, sering ditulis log x. Contoh:

1. Sederhanakan 24log31

log41

log 222 ++ !

Jawab:

)2431

41

log(24log31

log41

log 2222 xx=++

= 2log2 = 1

2. Sederhanakan!

a. 2 3log2

b. log2 98 c. log 150 + log 2 – log 3

Jawab:

a. 2 3log2

=3 b. log2 98 = 2 log (49. 2)

= 2 log ( 7. 7. 2) = 2 log 7 + 2 log 7 + 2 log 2 = 2 2 log 7 + 1

c. Log 150 + log 2 – log 3 = log ÷øö

çèæ

32.150

= log 100 = 2

3. Jika 5 log 3 = p tentukan 5 log 75!

Jawab: 5 log 75 = 5 log(3 x 5) = 5 log 3 + 5 log 25 = p + 5 log 5 2

= p + 2

Matematika X – Semester 1 SMAN 1 Bone-Bone 21

Latihan

Kerjakan soal-soal di bawah ini dengan benar ! 1. Nyatakan menjadi bentuk pangkat !

a. 2 log 21

2 = -21

b. 3 log 811

= -4

c. 25 log 0,2 = -21

d. 5 log 1 = 0 Jawab:

a. 2 log 21

2 = -21

2 ×××× = ….

b. 3 log 811

= -4

3 ×××× =…..

c. 25 log 0,2 = -21

25 ×××× = ….

d. 5 log 1 = 0 6 ×××× = ….

2. Hitung nilai logaritma berikut! a. 2 log 256

b. 3 log 243

1

Jawab: a. 2 log 256 = 2 log 2 ×××× = …. 2 log 2 = …. 1 = ….

b. 3 log 243

1 = 3 log 3 ××××

= …. 3 log 3 = …. 1 = …..

3. Tentukan nilai dari 5 log 3x 3 log 125! Jawab: 5 log 3x 3 log 125 = 5 log …. = 5 log …. = ….

4. Sederhanakan 32 log 8 !

Jawab : 32 log 8 = …. ×××× log 2 ××××

××××

= ××××××××

××××

××××

×××× log …. = … x … = ….

Matematika X – Semester 1 SMAN 1 Bone-Bone 22

Uji Kompetensi 3

A. Berilah tanda silang (x) huruf a, b, c, d, atau e pada jawaban yang paling benar!\

1. log4 64 + 3 log 81- 21

log 8 =….

a. 10 c. 7 e. 4

b. 9 d. 6

2. Jika log 2 = a, maka log 5 adalah ….

a. a c. 1 – a e. -1 b. 1 + 2 d. 3a

3. Jika log 2 = a, maka log 50 adalah ….

c. -1 c. 3a e. 2 - a

b. 2a d. 2a - 1

3. 4 2 log 5 = ….

d. 0,4 c. 1 e. 25

b. 0,2 d. 5

4. Jika log (2x + 6), mka x adalah…

a. 46 c. 48 e. 50

b. 47 d. 49

5. a log ÷øö

çèæ

b1

+ b log ÷øö

çèæ

c1

x c log ÷øö

çèæ

a1

=…..

a. -1 c. abc

1 e. 1 - abc

b. 1 d. 1 + abc

6. Bentuk sederhana dari 10 log 8 + 10 log 1,25 adalah….

a. 100 c. 3 e. 1 b. 10 d. 2

7. Jika 3 log 5 = p, maka nilai 5 log 3 adalah….

a. p4

c. p1

e. p4

1

b. p2

d. p2

1

Matematika X – Semester 1 SMAN 1 Bone-Bone 23

8. Nilai dari 3 log 251

. 5 log 8. 2 log 3 adalah….

a. -3 c. 1 e. 3 b. -2 d. 2

9. Jika 2 a = 3, maka 3 log 12 = ….

a. 2

2 a+ c.

aa

++

12

e. 2 + a1

b. a

a+2 d.

a1

1+

10. Jika 3 log 5 = p, maka nilai 3log5 adalah….

a. p4

1 c.

p1

e. p4

b. p2

1 d.

p2

11. Nilai x dari 5 log 0,2 = x adalah….

a. -2 c. 2 e. 4 b. -1 d. 3

12. Himpunan penyelesaian dari log 3x 2 + 2x – 4 = 0 adalah….

a. þýü

îíì- 1,

35

c. { }1 e. {-1}

b. þýü

îíì -- 1,

35

d. þýü

îíì

35

13. Nilai dari 2 27 log 2 81 adalah ….

a. 34

c. 42

e. 132

b. 43

d. 32

14. 33 log 27 sama dengan ….

a. 2 c. 6 e. 6

b. 3 d. 2

Matematika X – Semester 1 SMAN 1 Bone-Bone 24

B. Kerjakan soal-soal di bawah ini dengan benar !

1. Tentukan nilai dari log 30 - 10log

148 +

10log1

16 !

Jawab: .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... ....................................................................................................................................

2. Tentukan nilai dari persamaan berikut! a. log 2 + log 18 – log 6 + log 5 – log 3 b. 5 log 150 - 5 log 24 + 5 log 4 c. 3 log 81 + 3 log 243 - 3 log 27 Jawab: .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... ....................................................................................................................................

3. Hitunglah!

a. 4 log 256

b. Jika b log a = 6 dan b log c = 4, tentukan b log 4 3

3 2

c

a

Jawab: .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... ....................................................................................................................................

Matematika X – Semester 1 SMAN 1 Bone-Bone 25

4. Sederhanakan !

a. 52 32log b. 34 32log3

Jawab : .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... ....................................................................................................................................

5. Misal 2 log 3 = p dan 2 log 5 = q, nyatakan 2 log 225 dengan p dan q! Jawab: .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... .................................................................................................................................... ....................................................................................................................................