sifat - sifat logaritma yang sering kita pelajari terkadang hanya sekedar kita hafalkan saja tanpa mengetahui dari mana sifat tersebut berasal berikut saya sajikan slide dalam pembuktian masing2 sifat logaritma,.. untuk penjelasannya kalian dapat menyaksikan video di youtube... untuk penjelasan dari slide share ini dapat kalian simak videonya pada link berikut : https://youtu.be/JSU5gWgnrDU

1. PEMBUKTIAN SIFAT-SIFAT
LOGARITMA
Oleh :
Franxisca Kurniawati, S.Si.

2. Jika 𝒂 > 𝟎 dan 𝒂 ≠ 𝟏, 𝒎 > 𝟎 dan 𝒎 ≠ 𝟏, 𝒃 > 𝟎 , 𝒄 > 𝟎
Sifat-Sifat Logaritma
𝒂 𝒍𝒐𝒈 𝟏 = 𝟎, 𝒂 𝒍𝒐𝒈 𝒂 = 𝟏
𝒂
𝒍𝒐𝒈𝒃 + 𝒂
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈 (𝒃. 𝒄)
𝒂
𝒍𝒐𝒈𝒃 − 𝒂
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈
𝒃
𝒄
𝒂
𝒍𝒐𝒈 𝒃 𝒏
= 𝒏. 𝒂
𝒍𝒐𝒈 𝒃
𝒂 𝒎
𝒍𝒐𝒈 𝒃 =
𝟏
𝒎
. 𝒂 𝒍𝒐𝒈 𝒃
𝒂 𝒎
𝒍𝒐𝒈𝒃 𝒏 =
𝒏
𝒎
. 𝒂 𝒍𝒐𝒈𝒃
𝒂 𝒍𝒐𝒈
𝒃
𝒄
= − 𝒂 𝒍𝒐𝒈
𝒄
𝒃
𝒂 𝒍𝒐𝒈𝒃. 𝒃 𝒍𝒐𝒈 𝒄 = 𝒂 𝒍𝒐𝒈 𝒄
𝒂 𝒍𝒐𝒈𝒃 =
𝟏
𝒃 𝒍𝒐𝒈𝒂
𝒂
𝒂 𝒍𝒐𝒈𝒃 = 𝒃
𝒂 𝒍𝒐𝒈𝒃 =
𝒎 𝒍𝒐𝒈𝒃
𝒎 𝒍𝒐𝒈𝒂

3. 𝒂 𝒄 = 𝒃
↔ 𝒄 = 𝒂 𝒍𝒐𝒈 𝒃
𝒂
𝒍𝒐𝒈 𝒃 = 𝒄
↔ 𝒂 𝒄 = 𝒃
Ingat !!!

4. 𝒂 𝒍𝒐𝒈 𝟏 = 𝟎, 𝒂 𝒍𝒐𝒈 𝒂 = 𝟏
1

5. 𝒂
𝒍𝒐𝒈 𝟏 = 𝟎 𝒂
𝒍𝒐𝒈 𝒂 = 𝟏
𝑩𝒖𝒌𝒕𝒊 ∶
𝒂
𝒍𝒐𝒈 𝟏 = 𝟎
↔ 𝒂 𝟎
= 𝟏
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊
𝑩𝒖𝒌𝒕𝒊 ∶
𝒂
𝒍𝒐𝒈 𝒂 = 𝟏
↔ 𝒂 𝟏
= 𝒂
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊

6. 𝒂
𝒍𝒐𝒈𝒃 + 𝒂
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈 (𝒃. 𝒄)
2

7. 𝒂
𝒍𝒐𝒈 𝒃 + 𝒂
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈 (𝒃. 𝒄)
𝒂 𝒍𝒐𝒈 𝒃 = 𝒎
↔ 𝒂 𝒎 = 𝒃
𝒂 𝒍𝒐𝒈 𝒄 = 𝒏
↔ 𝒂 𝒏 = 𝒄
𝑩𝒖𝒌𝒕𝒊 ∶
𝒅𝒂𝒏𝒎𝒊𝒔𝒂𝒍:
𝒎𝒂𝒌𝒂 ∶ 𝒂
𝒍𝒐𝒈 𝒃 + 𝒂
𝒍𝒐𝒈 𝒄 = 𝒎 + 𝒏
= 𝒂
𝒍𝒐𝒈 𝒂(𝒎+𝒏)
= 𝒂
𝒍𝒐𝒈 (𝒂 𝒎
. 𝒂 𝒏
)
= 𝒂
𝒍𝒐𝒈 (𝒃. 𝒄)
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊
𝒂 𝒍𝒐𝒈 𝒙 = (𝒎 + 𝒏)
↔ 𝒂 𝒎+𝒏 = 𝒙

8. 𝒂
𝒍𝒐𝒈𝒃 − 𝒂
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈
𝒃
𝒄
3

9. 𝒂
𝒍𝒐𝒈 𝒃 − 𝒂
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈
𝒃
𝒄
𝒂
𝒍𝒐𝒈 𝒃 = 𝒎
↔ 𝒂 𝒎 = 𝒃
𝒂
𝒍𝒐𝒈 𝒄 = 𝒏
↔ 𝒂 𝒏 = 𝒄
𝑩𝒖𝒌𝒕𝒊 ∶
𝒅𝒂𝒏𝒎𝒊𝒔𝒂𝒍:
𝒎𝒂𝒌𝒂 ∶ 𝒂
𝒍𝒐𝒈 𝒃 − 𝒂
𝒍𝒐𝒈 𝒄 = 𝒎 − 𝒏
= 𝒂
𝒍𝒐𝒈 𝒂(𝒎−𝒏)
= 𝒂
𝒍𝒐𝒈
𝒂 𝒎
𝒂 𝒏
= 𝒂
𝒍𝒐𝒈
𝒃
𝒄
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊
𝒂 𝒍𝒐𝒈 𝒙 = (𝒎 − 𝒏)
↔ 𝒂 𝒎−𝒏 = 𝒙

10. 𝒂
𝒍𝒐𝒈 𝒃 𝒏
= 𝒏. 𝒂
𝒍𝒐𝒈 𝒃
4

11. 𝒂
𝒍𝒐𝒈 𝒃 𝒏
= 𝒏. 𝒂
𝒍𝒐𝒈 𝒃
𝑩𝒖𝒌𝒕𝒊 ∶
𝒂
𝒍𝒐𝒈 𝒃 𝒏
= 𝒂
𝒍𝒐𝒈 (𝒃 × 𝒃 × … × 𝒃)
= 𝒂 𝒍𝒐𝒈 𝒃 + 𝒂 𝒍𝒐𝒈 𝒃 + ⋯ + 𝒂 𝒍𝒐𝒈 𝒃
= 𝒏. 𝒂
𝒍𝒐𝒈 𝒃
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊
𝒔𝒆𝒃𝒂𝒏𝒚𝒂𝒌 𝒏 𝒇𝒂𝒌𝒕𝒐𝒓
#2
𝒔𝒆𝒃𝒂𝒏𝒚𝒂𝒌 𝒏 𝒇𝒂𝒌𝒕𝒐𝒓
𝒂
𝒍𝒐𝒈𝒃 + 𝒂
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈 (𝒃. 𝒄)

12. 𝒂 𝒎
𝒍𝒐𝒈 𝒃 =
𝟏
𝒎
. 𝒂 𝒍𝒐𝒈 𝒃
5

13. 𝒂 𝒎
𝒍𝒐𝒈 𝒃 =
𝟏
𝒎
. 𝒂
𝒍𝒐𝒈 𝒃
𝒂 𝒎
𝒍𝒐𝒈 𝒃 = 𝒙
↔ 𝒂 𝒎𝒙
= 𝒃
𝑩𝒖𝒌𝒕𝒊 ∶
𝒎𝒊𝒔𝒂𝒍:
𝒎𝒂𝒌𝒂 ∶ 𝒂 𝒎𝒙 = 𝒃
↔ 𝒎𝒙 = 𝒂 𝒍𝒐𝒈 𝒃
𝒙 =
𝒂
𝒍𝒐𝒈 𝒃
𝒎
𝒂 𝒎
𝒍𝒐𝒈 𝒃 =
𝟏
𝒎
. 𝒂
𝒍𝒐𝒈 𝒃
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊

14. 𝒂 𝒎
𝒍𝒐𝒈𝒃 𝒏 =
𝒏
𝒎
. 𝒂 𝒍𝒐𝒈𝒃
6

15. 𝒂 𝒎
𝒍𝒐𝒈 𝒃 𝒏 =
𝒏
𝒎
. 𝒂 𝒍𝒐𝒈 𝒃
𝑩𝒖𝒌𝒕𝒊 ∶
𝒂 𝒎
𝒍𝒐𝒈 𝒃 𝒏 =
𝒏
𝒎
. 𝒂 𝒍𝒐𝒈 𝒃
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊
𝒂
𝒍𝒐𝒈 𝒃 𝒏
= 𝒏. 𝒂
𝒍𝒐𝒈 𝒃 𝒂 𝒎
𝒍𝒐𝒈 𝒃 =
𝟏
𝒎
. 𝒂
𝒍𝒐𝒈 𝒃
#4 #5
𝒅𝒂𝒏

16. 𝒂
𝒍𝒐𝒈
𝒃
𝒄
= − 𝒂
𝒍𝒐𝒈
𝒄
𝒃
7

17. 𝒂
𝒍𝒐𝒈
𝒃
𝒄
= − 𝒂
𝒍𝒐𝒈
𝒄
𝒃
𝑩𝒖𝒌𝒕𝒊 ∶
𝒂 𝒍𝒐𝒈
𝒃
𝒄
= 𝒂 𝒍𝒐𝒈
𝒄
𝒃
−𝟏
= −𝟏. 𝒂 𝒍𝒐𝒈
𝒄
𝒃
= − 𝒂
𝒍𝒐𝒈
𝒄
𝒃
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊
𝒂 𝒍𝒐𝒈 𝒃 𝒏 = 𝒏. 𝒂 𝒍𝒐𝒈 𝒃
#4

18. 𝒂
𝒍𝒐𝒈𝒃. 𝒃
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈 𝒄
8

19. 𝒂
𝒍𝒐𝒈𝒃. 𝒃
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈 𝒄
𝒂 𝒍𝒐𝒈 𝒃 = 𝒙
↔ 𝒂 𝒙 = 𝒃
𝑩𝒖𝒌𝒕𝒊 ∶
𝒎𝒊𝒔𝒂𝒍:
𝒎𝒂𝒌𝒂 ∶ 𝒂 𝒍𝒐𝒈𝒃. 𝒃 𝒍𝒐𝒈 𝒄 = 𝒙. 𝒂 𝒙
𝒍𝒐𝒈 𝒄
=
𝒙
𝒙
. 𝒂 𝒍𝒐𝒈 𝒄
= 𝒂 𝒍𝒐𝒈 𝒄
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊
𝒂 𝒎
𝒍𝒐𝒈 𝒃 =
𝟏
𝒎
. 𝒂
𝒍𝒐𝒈 𝒃
#5

20. 𝒂 𝒍𝒐𝒈𝒃 =
𝟏
𝒃 𝒍𝒐𝒈𝒂
9

21. 𝒂
𝒍𝒐𝒈𝒃 =
𝟏
𝒃 𝒍𝒐𝒈𝒂
𝒂 𝒍𝒐𝒈 𝒃 = 𝒙
↔ 𝒂 𝒙 = 𝒃
𝑩𝒖𝒌𝒕𝒊 ∶
𝒎𝒊𝒔𝒂𝒍:
𝒎𝒂𝒌𝒂 ∶
𝒂 𝒙
𝟏
𝒙 = 𝒃
𝟏
𝒙
𝒂 = 𝒃
𝟏
𝒙
↔ 𝒃
𝒍𝒐𝒈 𝒂 =
𝟏
𝒙
𝒙 =
𝟏
𝒃 𝒍𝒐𝒈 𝒂
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊

22. 𝒂
𝒂 𝒍𝒐𝒈𝒃
= 𝒃
10

23. 𝒂
𝒂 𝒍𝒐𝒈𝒃
= 𝒃
𝒂 𝒍𝒐𝒈 𝒃 = 𝒙
↔ 𝒂 𝒙 = 𝒃
𝑩𝒖𝒌𝒕𝒊 ∶
𝒎𝒊𝒔𝒂𝒍:
𝒎𝒂𝒌𝒂 ∶
𝒂
𝒂 𝒍𝒐𝒈𝒃
= 𝒂 𝒙
= 𝒃
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊

24. 𝒂 𝒍𝒐𝒈𝒃 =
𝒎 𝒍𝒐𝒈𝒃
𝒎 𝒍𝒐𝒈𝒂
11

25. 𝒂
𝒍𝒐𝒈𝒃 =
𝒎
𝒍𝒐𝒈𝒃
𝒎 𝒍𝒐𝒈𝒂
𝑩𝒖𝒌𝒕𝒊 ∶
𝒂
𝒍𝒐𝒈 𝒃 = 𝒂
𝒍𝒐𝒈 𝒎 . 𝒎
𝒍𝒐𝒈 𝒃
=
𝟏
𝒎 𝒍𝒐𝒈𝒂
. 𝒎
𝒍𝒐𝒈 𝒃
=
𝒎
𝒍𝒐𝒈𝒃
𝒎 𝒍𝒐𝒈𝒂
𝒕𝒆𝒓𝒃𝒖𝒌𝒕𝒊
𝒂
𝒍𝒐𝒈𝒃. 𝒃
𝒍𝒐𝒈 𝒄 = 𝒂
𝒍𝒐𝒈 𝒄
𝒂 𝒍𝒐𝒈𝒃 =
𝟏
𝒃 𝒍𝒐𝒈𝒂
#9
#8