Phương pháp giải phương trình bất phương trình chứa logarit và các bài toán liên quan_2

Header Page 1 of 128.

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

▲Þ ❱❿◆ ✣Ù❈

P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍
❇❻❚ P❍×❒◆● ❚❘➐◆❍ ❈❍Ù❆ ▲❖●❆❘■❚
❱⑨ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ▲■➊◆ ◗❯❆◆

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆ ✲ ◆❿▼ ✷✵✶✹

Footer Page 1 of 128.


Header Page 2 of 128.

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

▲Þ ❱❿◆ ✣Ù❈

P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍
❇❻❚ P❍×❒◆● ❚❘➐◆❍ ❈❍Ù❆ ▲❖●❆❘■❚
❱⑨ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ▲■➊◆ ◗❯❆◆

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈
❈❤✉②➯♥ ♥❣❤➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P

▼➣ sè ✻✵✳✹✻✳✵✶✳✶✸
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝

●❙✳ ❚❙❑❍✳ ◆●❯❨➍◆ ❱❿◆ ▼❾❯
❚❍⑩■ ◆●❯❨➊◆ ✲ ◆❿▼ ✷✵✶✹

Footer Page 2 of 128.


Header Page 3 of 128.

▼ö❝ ❧ö❝
▼ð ✤➛✉
✸
✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t ✈➔ ❝→❝ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✺
✶✳✶

❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✷

❈→❝ ✤à♥❤ ❧þ ❜ê trñ

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✸

▲î♣ ❤➔♠ t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳

✶✵

✶✳✸✳✶

▲î♣ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✸✳✷

▲î♣ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✷ P❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛
❧♦❣❛r✐t
✶✸
✷✳✶

✷✳✷

✷✳✸

P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t

✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✷✳✶✳✶

P❤÷ì♥❣ ♣❤→♣ ♠ô ❤â❛ ✈➔ ✤÷❛ ✈➲ ❝ò♥❣ ❝ì sè ✳ ✳ ✳ ✳

✶✸

✷✳✶✳✷

P❤÷ì♥❣ ♣❤→♣ ✤➦t ➞♥ ♣❤ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✷✳✶✳✸

P❤÷ì♥❣ ♣❤→♣ ❤➡♥❣ sè ❜✐➳♥ t❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✷✳✶✳✹

P❤÷ì♥❣ ♣❤→♣ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✷✳✶✳✺

Ù♥❣ ❞ö♥❣ ✤à♥❤ ❧þ ▲❛❣r❛♥❣❡✱ ✤à♥❤ ❧þ ❘♦❧❧❡

✳ ✳ ✳ ✳

✷✾

✷✳✶✳✻

P❤÷ì♥❣ ♣❤→♣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✸

✷✳✶✳✼

P❤÷ì♥❣ ♣❤→♣ ✤→♥❤ ❣✐→ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺

P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳

✸✻

✷✳✷✳✶

P❤÷ì♥❣ ♣❤→♣ ♠ô ❤â❛ ✈➔ ✤÷❛ ✈➲ ❝ò♥❣ ❝ì sè ✳ ✳ ✳ ✳

✸✻

✷✳✷✳✷

P❤÷ì♥❣ ♣❤→♣ ✤➦t ➞♥ ♣❤ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

✷✳✷✳✸

P❤÷ì♥❣ ♣❤→♣ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

✷✳✷✳✹

P❤÷ì♥❣ ♣❤→♣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✹

✷✳✷✳✺

P❤÷ì♥❣ ♣❤→♣ ✤→♥❤ ❣✐→ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺

P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♠ët sè ❤➺ ❝❤ù❛ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

✐

Footer Page 3 of 128.


Header Page 4 of 128.

✷✳✸✳✶

P❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ t÷ì♥❣ ✤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

✷✳✸✳✷

P❤÷ì♥❣ ♣❤→♣ ✤➦t ➞♥ ♣❤ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽

✷✳✸✳✸

P❤÷ì♥❣ ♣❤→♣ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾

✷✳✸✳✹

P❤÷ì♥❣ ♣❤→♣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✶

✷✳✸✳✺

P❤÷ì♥❣ ♣❤→♣ ✤→♥❤ ❣✐→ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✸

✸ ❈→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ sè ❧♦❣❛r✐t
✸✳✶

P❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr♦♥❣ ❧î♣ ❤➔♠
❧♦❣❛r✐t

✸✳✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✻

✸✳✶✳✶

P❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr♦♥❣ ❧î♣ ❤➔♠ ❧♦❣❛r✐t

✳ ✳ ✳ ✳ ✳

✺✻

✸✳✶✳✷

❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr♦♥❣ ❧î♣ ❤➔♠ ❧♦❣❛r✐t ✳ ✳ ✳

✻✹

❈→❝ ❜➔✐ t♦→♥ ✈➲ ❞➣② sè ✈➔ ❣✐î✐ ❤↕♥ ❞➣② sè s✐♥❤ ❜ð✐ ❤➔♠
❧♦❣❛r✐t

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✻✼

✼✺
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐✐

Footer Page 4 of 128.

✺✻

✼✻


Header Page 5 of 128.

▼ð ✤➛✉
P❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝ì ❜↔♥
✈➔ q✉❛♥ trå♥❣ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❜➟❝ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣✳
✣➦❝ ❜✐➺t ❧➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ❧➔ ♥❤ú♥❣
♥ë✐ ❞✉♥❣ ❤❛② ✈➔ ❦❤â ✤è✐ ✈î✐ ❤å❝ s✐♥❤ ✈➔ ❝❤ó♥❣ t❤÷í♥❣ ①✉➜t ❤✐➺♥ tr♦♥❣
❝→❝ ✤➲ t❤✐ t✉②➸♥ s✐♥❤ ✤↕✐ ❤å❝✱ ❝❛♦ ✤➥♥❣ ✈➔ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐✳ ❱✐➺❝
❣✐↔♥❣ ❞↕② ❤➔♠ sè ❧♦❣❛r✐t ✤➣ ✤÷ñ❝ ✤÷❛ ✈➔♦ ❝❤÷ì♥❣ tr➻♥❤ ❧î♣ ✶✷ tr♦♥❣ ✤â
♣❤➛♥ ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ❝❤✐➳♠
✈❛✐ trá trå♥❣ t➙♠✳

❚✉② ♥❤✐➯♥ ❞♦ t❤í✐ ❣✐❛♥ ❤↕♥ ❤➭♣ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ ♣❤ê t❤æ♥❣ ♥➯♥ tr♦♥❣
s→❝❤ ❣✐→♦ ❦❤♦❛ ❦❤æ♥❣ ♥➯✉ ✤÷ñ❝ ✤➛② ✤õ ✈➔ ❝❤✐ t✐➳t t➜t ❝↔ ❝→❝ ❞↕♥❣ ❜➔✐
t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥
q✉❛♥✳ ❱➻ ✈➟② ❤å❝ s✐♥❤ t❤÷í♥❣ ❣➦♣ ♥❤✐➲✉ ❦❤â ❦❤➠♥ ❦❤✐ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥
♥➙♥❣ ❝❛♦ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t tr♦♥❣ ❝→❝ ✤➲
t❤✐ ✤↕✐ ❤å❝✱ ❝❛♦ ✤➥♥❣ ✈➔ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ t➔✐ ❧✐➺✉
t❤❛♠ ❦❤↔♦ ✈➲ ❧♦❣❛r✐t ✈î✐ ♥ë✐ ❞✉♥❣ ❦❤→❝ ♥❤❛✉ ♥❤÷♥❣ ❝❤÷❛ ❝â ❝❤✉②➯♥ ✤➲
r✐➯♥❣ ❦❤↔♦ s→t ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ♠ët ❝→❝❤
❤➺ t❤è♥❣✳
✣➦❝ ❜✐➺t✱ ♥❤✐➲✉ ❞↕♥❣ t♦→♥ ✈➲ ✤↕✐ sè ✈➔ ❧♦❣❛r✐t ❝â q✉❛♥ ❤➺ ❝❤➦t ❝❤➩
✈î✐ ♥❤❛✉✱ ❦❤æ♥❣ t❤➸ t→❝❤ rí✐ ✤÷ñ❝✳ ◆❤✐➲✉ ❜➔✐ t♦→♥ ❝❤ù❛ ❧♦❣❛r✐t ❝➛♥ ❝â
sü trñ ❣✐ó♣ ❝õ❛ ✤↕✐ sè✱ ❣✐↔✐ t➼❝❤ ✈➔ ♥❣÷ñ❝ ❧↕✐✳
❉♦ ✤â✱ ✤➸ ✤→♣ ù♥❣ ♥❤✉ ❝➛✉ ✈➲ ❣✐↔♥❣ ❞↕②✱ ❤å❝ t➟♣ ✈➔ ❣â♣ ♣❤➛♥ ♥❤ä
❜➨ ✈➔♦ sü ♥❣❤✐➺♣ ❣✐→♦ ❞ö❝✱ ❧✉➟♥ ✈➠♥ ✧P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱
❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✧ ♥❤➡♠ ❤➺ t❤è♥❣
❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ❦➳t

✸

Footer Page 5 of 128.


Header Page 6 of 128.

ủ ợ tự số t tờ ủ ồ ồ
ữỡ ữỡ tr t ữỡ tr ự rt
ỹ ởt số ợ t ợ
ữủ ữỡ
ữỡ t ừ số rt tự q
t t ừ số rt

ỵ ờ trủ
ợ t t t
ữỡ Pữỡ tr t ữỡ tr ữỡ tr ự
rt
r ữỡ ữỡ tr ự rt
r ữỡ t ữỡ tr ự rt
r ữỡ ởt số ự rt
ữỡ t q số rt
ữỡ tr t ữỡ tr tr ợ rt
t số ợ số s rt
tọ ỏ t ỡ s s ố ợ sữ s
ồ ữớ t trỹ t ữợ t
tr t ỳ ự tổ
ổ t ỡ t ổ tr
ỏ t trữớ ồ ồ ồ rữớ
P ỗ ú ù t tổ
t


ỵ ự



Footer Page 6 of 128.


Header Page 7 of 128.

❈❤÷ì♥❣ ✶
❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t ✈➔ ❝→❝

❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥
✶✳✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t
f (x) = loga x, 0 < a = 1 ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ sè ❧♦❣❛r✐t ❝ì sè a✳
◆❤➟♥ ①➨t r➡♥❣ t➟♣ ①→❝ ✤à♥❤ D = (0; +∞) ✈➔ t➟♣ ❣✐→ trà I = R✳
❚r♦♥❣ ❝→❝ ♣❤➛♥ t✐➳♣ t❤❡♦✱ t❛ ❣✐↔ sû 0 < a = 1✳
◆❤➟♥ ①➨t r➡♥❣ ❤➔♠ sè f (x) = loga x ❧✐➯♥ tö❝ ✈➔ ❝â ✤↕♦ ❤➔♠ ✈î✐ ♠å✐
x > 0✱ ❤ì♥ ♥ú❛
1
f (x) =
.
x ln a
❚❛ ❦❤↔♦ s→t t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè f (x) = loga x tr♦♥❣ ✷ tr÷í♥❣ ❤ñ♣✳
✲ ❚r÷í♥❣ ❤ñ♣ ✶✿ a > 1✳
❑❤✐ ✤â✱ ln a > 0 ♥➯♥ s✉② r❛
1
f (x) =
> 0, ∀x > 0.
x ln a
❱➟②✱ ❦❤✐ a > 1 t❤➻ f (x) = loga x ❧➔ ❤➔♠ ✤ç♥❣ ❜✐➳♥ tr➯♥ ❉✳
❚❛ ❧↕✐ ❝â f (1) = 0, f (a) = 1 ✈➔ lim loga x = −∞; lim loga x = +∞.
+
❍➔♠ sè

x→+∞

x→0

❚❛ ❝â ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ s❛✉✿

x

0

1

1

y = loga x

0
−∞

✺

Footer Page 7 of 128.

a

+∞
+∞


Header Page 8 of 128.

0 < a < 1✳
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② f (x) < 0, ∀x ∈ D ✳
❱➟②✱ ❦❤✐ 0 < a < 1 t❤➻ f (x) = loga x ❧➔ ❤➔♠

✲ ❚r÷í♥❣ ❤ñ♣ ✷✿

sè ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ❉✳

❚❛ ❝â ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ s❛✉✿

x

0
+∞

1

a

+∞

0

y = loga x

1
−∞

❚➼♥❤ ❝❤➜t ✶✳✶✳ f (x) = logax
✈➔ ♥❣❤à❝❤ ❜✐➳♥ ❦❤✐

❧➔ ❤➔♠ ✤ç♥❣ ❜✐➳♥ tr➯♥

D = R+

❦❤✐

a>1

0 < a < 1.

❚➼♥❤ ❝❤➜t ✶✳✷✳

❱î✐ ♠å✐

❚➼♥❤ ❝❤➜t ✶✳✸✳

❱î✐ ♠å✐

a > 0✱ a = 1 ✈➔ x1 , x2 ∈ (0; +∞)✱
x1
loga (x1 x2 ) = loga x1 + loga x2 ✱ loga = loga x1 − loga x2 .
x2
a > 0✱ a = 1

✈➔

x > 0✳

❱î✐

α

t❛ ❝â

❜➜t ❦ý✱ t❛ ❝â

loga xα = αloga x.

❚➼♥❤ ❝❤➜t ✶✳✹✳

❱î✐ ♠å✐

0 < a = 1, 0 < c = 1
loga x =

❚➼♥❤ ❝❤➜t ✶✳✺✳

❍➔♠ sè

❚➼♥❤ ❝❤➜t ✶✳✻✳

❱î✐ ♠å✐

✈➔

x > 0✱

t❛ ❝â

logc x
.
logc x

f (x) = loga x (0 < a = 1) ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐
1

✤✐➸♠ x ∈ (0; +∞) ✈➔ (loga x) =
. ◆➳✉ ❤➔♠ sè u = u(x) ❝â ✤↕♦ ❤➔♠
x ln a
tr➯♥ ❦❤♦↔♥❣ J ∈ R t❤➻ ❤➔♠ sè y = loga u(x)✱ (0 < a = 1) ❝â ✤↕♦ ❤➔♠
u (x)
tr➯♥ ❏ ✈➔ (loga u(x)) =
.
u(x) ln a

•

❑❤✐

a>1

•

❑❤✐

0
t❤➻

a > 0✱ a = 1

x1 , x2 ∈ (0; +∞)✱

loga x1 < loga x2 ⇔ x1 < x2 .
t❤➻

loga x1 < loga x2 ⇔ x1 > x2 .

✻

Footer Page 8 of 128.

✈➔

t❛ ❝â


Header Page 9 of 128.

✶✳✷ ❈→❝ ✤à♥❤ ❧þ ❜ê trñ

✣à♥❤ ❧➼ ✶✳✶✳

c ∈ (a; b)

t❤➻ tç♥ t↕✐

✣à♥❤ ❧➼ ✶✳✷✳

y = f (x) ❧✐➯♥
❝❤♦ f (c) = 0✳

◆➳✉ ❤➔♠ sè
s❛♦

◆➳✉ ❤➔♠ sè

tö❝ tr➯♥

[a; b]

✈➔

f (a).f (b) < 0

y = f (x) ❧✐➯♥ tö❝ tr➯♥ [a; b]✱ f (a) = A, f (b) = B

t❤➻ ❤➔♠ sè ♥❤➟♥ ♠å✐ ❣✐→ trà tr✉♥❣ ❣✐❛♥ ❣✐ú❛ ❆ ✈➔ ❇✳

❍➺ q✉↔ ✶✳✶✳

◆➳✉ ❤➔♠ sè

y = f (x)

❧✐➯♥ tö❝ tr➯♥

[a; b]

t❤➻ ♥â ♥❤➟♥ ♠å✐

❣✐→ trà tr✉♥❣ ❣✐❛♥ ❣✐ú❛ ❣✐→ trà ❧î♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t✳

✣à♥❤ ❧➼ ✶✳✸
[a; b]✱

✳

❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣

s❛♦ ❝❤♦

f : [a; b] → R t❤ä❛ ♠➣♥ ❢
(a; b) ✈➔ f (a) = f (b) t❤➻ tç♥

✭❘♦❧❧❡✮ ❈❤♦ ❤➔♠ sè

❧✐➯♥ tö❝ tr➯♥
t↕✐

c ∈ (a; b)

f (c) = 0✳

❈❤ù♥❣ ♠✐♥❤✳
❱➻

f (x) ❧✐➯♥ tö❝

tr➯♥

[a; b] ♥➯♥ t❤❡♦

✤à♥❤ ❧➼ ❲❡✐❡rstr❛ss

f (x) ♥❤➟♥ ❣✐→

trà

[a; b]✳
tr➯♥ [a; b]✱

❧î♥ ♥❤➜t ▼ ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t ♠ tr➯♥

M = m t❛ ❝â f (x) ❧➔ ❤➔♠ ❤➡♥❣
❞♦ ✤â ✈î✐ ♠å✐ c ∈ (a; b)
❧✉æ♥ ❝â f (c) = 0✳
✲ ❑❤✐ M > m✱ ✈➻ f (a) = f (b) ♥➯♥ tç♥ t↕✐ c ∈ (a; b) s❛♦ ❝❤♦ f (c) = m
❤♦➦❝ f (c) = M ✱ t❤❡♦ ❜ê ✤➲ ❋❡r♠❛t s✉② r❛ f (c) = 0✳
✲ ❑❤✐

❍➺ q✉↔ ✶✳✷✳

◆➳✉ ❤➔♠ sè

f (x)

❝â ♥ ♥❣❤✐➺♠ ✭♥ ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ❧î♥ ❤ì♥ ✶✮ tr➯♥
♥❤➜t

n−1

(a; b)✳

♥❣❤✐➺♠ tr➯♥

❍➺ q✉↔ ✶✳✸✳

❍➺ q✉↔ ✶✳✹✳

f (x) ❝â ✤↕♦ ❤➔♠
f (x) ❝â ♥❤✐➲✉ ♥❤➜t

◆➳✉ ❤➔♠ sè

✈æ ♥❣❤✐➺♠ tr➯♥

(a; b)

(a; b) ✈➔ f (x)
(a; b) t❤➻ f (x) ❝â ➼t

❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣

t❤➻

◆➳✉ ❤➔♠ sè

f (x)

✶ ♥❣❤✐➺♠

n+1

✣à♥❤ ❧➼ ✶✳✹
tr➯♥ ✤♦↕♥

♥❣❤✐➺♠ tr➯♥

✳

✭▲❛❣r❛♥❣❡✮

[a; b]✱

(a; b)✳
f : [a; b] → R t❤ä❛ ♠➣♥
(a; b)✱ ❦❤✐ ✤â ∃c ∈ (a; b) :

❈❤♦ ❤➔♠ sè

❦❤↔ ✈✐ tr➯♥ ❦❤♦↔♥❣

f (c) =

f (b) − f (a)
.
b−a
✼

Footer Page 9 of 128.

(a; b) ✈➔ f (x)
(a; b) t❤➻ f (x) ❝â

❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣

❝â ♥❤✐➲✉ ♥❤➜t ♥ ♥❣❤✐➺♠ ✭♥ ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✮ tr➯♥
♥❤✐➲✉ ♥❤➜t

(a; b) ✈➔ f (x)
tr➯♥ (a; b)✳

tr➯♥ ❦❤♦↔♥❣

❢ ❧✐➯♥ tö❝


Header Page 10 of 128.

❈❤ù♥❣ ♠✐♥❤✳
❳➨t ❤➔♠ sè

F (x) = f (x) −

f (b) − f (a)
x.
b−a

❚❛ ❝â

F (x) ❧➔ ❤➔♠ ❧✐➯♥
F (a) = F (b)✳

tö❝ tr➯♥ ✤♦↕♥

[a; b]

✱ ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣

(a; b)

✈➔

c ∈ (a; b) s❛♦ ❝❤♦ F (c) = 0✳
f (b) − f (a)
f (b) − f (a)
F (x) = f (x) −
✱ s✉② r❛ f (c) =
✳
b−a
b−a

❚❤❡♦ ✤à♥❤ ❧➼ ❘♦❧❧❡ tç♥ t↕✐
▼➔

❍➺ q✉↔ ✶✳✺✳

◆➳✉

F (x) = 0 ✈î✐ ♠å✐ ① t❤✉ë❝ ❦❤♦↔♥❣ (a; b) t❤➻ F (x) ❜➡♥❣

❤➡♥❣ sè tr➯♥ ❦❤♦↔♥❣ ✤â✳

✣à♥❤ ❧➼ ✶✳✺✳
✲ ◆➳✉

✲ ◆➳✉

f (x) ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (a; b)✳
f (x) > 0, ∀x ∈ (a; b) t❤➻ f (x) ✤ç♥❣ ❜✐➳♥ tr➯♥ (a; b)✳
f (x) < 0, ∀x ∈ (a; b) t❤➻ f (x) ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ (a; b)✳

✣à♥❤ ❧➼ ✶✳✻
❦ý

❈❤♦ ❤➔♠ sè

✳

✭❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③✮

a1 , a2 , ..., an

✈➔

b1 , b2 , ..., bn ✳

❈❤♦ ❤❛✐ ❝➦♣ ❞➣② sè ❜➜t

❑❤✐ ✤â

(a1 b1 + a2 b2 + ... + an bn )2 ≤ (a21 + a22 + ... + a2n )(b21 + b22 + ... + b2n ).
❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

❈❤ù♥❣ ♠✐♥❤✳

∃k

✤➸

ai = kbi , ∀i ∈ (1, 2, ..., n).

❳➨t t❛♠ t❤ù❝ ❜➟❝ ❤❛✐

f (x) = (a21 +a22 +...+a2n )x2 −2(a1 b1 +a2 b2 +...+an bn )x+(b21 +b22 +...+b2n )
✲ ◆➳✉

a21 + a22 + ... + a2n = 0 ⇔ a1 = a2 = ... = an = 0

❜➜t ✤➥♥❣ t❤ù❝ ❤✐➸♥

♥❤✐➯♥ ✤ó♥❣✳
✲ ◆➳✉

a21 + a22 + ... + a2n > 0✱

t❛ ✈✐➳t

f (x)

❞÷î✐ ❞↕♥❣

f (x) = (a1 x − b1 )2 + (a2 x − b2 )2 + ... + (an x − bn )2 ≥ 0, ∀x ∈ R.
❚❤❡♦ ✤à♥❤ ❧þ ✈➲ ❞➜✉ ❝õ❛ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ t❤➻

∆ = (a1 b1 + a2 b2 + ... + an bn )2 − (a21 + a22 + ... + a2n )(b21 + b22 + ... + b2n ) ≤ 0

⇔ (a1 b1 + a2 b2 + ... + an bn )2 ≤ (a21 + a22 + ... + a2n )(b21 + b22 + ... + b2n ).
✽

Footer Page 10 of 128.


Header Page 11 of 128.

❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

∃k

✤➸



 a1 x − b 1 = 0
a2 x − b 2 = 0

 ..................
an x − bn = 0

❤❛② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

ai = kbi , ∀i ∈ (1, 2, ..., n).

✣à♥❤ ❧➼ ✶✳✼

✳

✭❇➜t ✤➥♥❣ t❤ù❝ ❇❡r♥♦✉❧❧✐✮

(1 + x)α ≤ 1 + αx
(1 + x)α ≥ 1 + αx

❦❤✐
❦❤✐

●✐↔ sû

x > −1✳

❑❤✐ ✤â

0≤α≤1
α ≤ 0 ∨ α ≥ 1.

❈❤ù♥❣ ♠✐♥❤✳

α = 0 ❤♦➦❝ α = 1 t❤➻ t❛ t❤✉ ✤÷ñ❝ ✤➥♥❣ t❤ù❝✳
α
✲ ❑❤✐ α < 0 ❤♦➦❝ α > 1✱ ①➨t ❤➔♠ sè f (x) = (1 + x) − αx − 1✱
x > −1✳
α−1
❚❛ ❝â f (x) = α(1 + x)
− α = α (1 + x)α−1 − 1 ✈➔

✲ ❑❤✐

✈î✐

f (x) = 0 ⇔ x = 0✱ ♥➯♥ f (x) ≥ 0, ∀x > −1 ⇔ (1 + x)α ≥ 1 + αx,
∀x > −1.
✲ ❑❤✐ 0 < α < 1✱ ①➨t ❤➔♠ sè f (x) ♥❤÷ tr➯♥✱ t❛ ❝â
f (x) ≤ 0, ∀x > −1 ⇔ (1 + x)α ≤ 1 + αx, ∀x > −1.

◆❤➟♥ ①➨t ✶✳✶✳

❑❤✐ t❤❛②

x

❜ð✐

x−1

xα + (1 − x)α ≤ 1
xα + (1 − x)α ≥ 1

❱➼ ❞ö ✶✳✶✳

❈❤♦

m>0

t❛ ❝â

❦❤✐
❦❤✐

0≤α≤1
α ≤ 0 ∨ α ≥ 1.

❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ❝á♥

a, b, c

❧➔ ✸ sè t❤ü❝ s❛♦

❝❤♦

a
b
c
+
+
=0
m+2 m+1 m
2
❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❦❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ax + bx + c = 0
♥❣❤✐➺♠ tr♦♥❣ ❦❤♦↔♥❣ (0; 1)✳
●✐↔✐✳
❳➨t ❤➔♠ sè

a
b
c
xm+2 +
xm+1 + xm
m+2

m+1
m
❧✐➯♥ tö❝ tr➯♥ [0; 1]✳ ❑❤↔ ✈✐ tr♦♥❣ (0; 1) ✈➔
F (x) = xm−1 (ax2 + bx + c).
F (x) =

✾

Footer Page 11 of 128.

❝â ➼t ♥❤➜t ✶


Header Page 12 of 128.

◆❣♦➔✐ r❛

F (0) = F (1) = 0.

⑩♣ ❞ö♥❣ ✤à♥❤ ❧þ ❘♦❧❧❡ ❦❤✐ ✤â

∃α ∈ (0; 1)

s❛♦ ❝❤♦

F (α) = 0
⇔ αm−1 (aα2 + bα + c) = 0
⇔ aα2 + bα + c = 0.

❱➟② ♣❤÷ì♥❣ tr➻♥❤

ax2 + bx + c = 0

❝â ♥❣❤✐➺♠

α ∈ (0; 1)✳

✭✤✐➲✉ ♣❤↔✐

❝❤ù♥❣ ♠✐♥❤✮

❱➼ ❞ö ✶✳✷✳

❈❤ù♥❣ ♠✐♥❤ r➡♥❣

2014
1
1
< ln
<
.
2014
2013 2013
●✐↔✐✳

f (x) = ln x✱ t❛ ❝â
1
f (x) = .
x
❍➔♠ f (x) ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [2013; 2014]✱ ❝â ✤↕♦ ❤➔♠ tr➯♥ (2013; 2014)✳

❚❤❡♦ ✤à♥❤ ❧þ ❧❛❣r❛♥❣❡ tç♥ t❛✐ c ∈ (2013; 2014) s❛♦ ❝❤♦
❳➨t ❤➔♠ sè

f (2014) − f (2013)
1
= f (c) ⇔ ln 2014 − ln 2013 =
2014 − 2013
c
2014 1
⇔ ln
= .
2013
c
c ∈ (2013; 2014) t❛
1
1
1
< <
.
2014
c
2013

❱î✐

❝â

❙✉② r❛

1

2014
1
< ln
<
.
2014
2013 2013

❱➟②✱ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

✶✳✸ ▲î♣ ❤➔♠ t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥
t➼♥❤

✶✳✸✳✶ ▲î♣ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳
f (x)
❍➔♠ sè

❝❤✉ ❦ý

a (a ∈
/ {0, 1, −1})

✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤

tr➯♥ ▼ ♥➳✉

✶✵

Footer Page 12 of 128.

M ⊂ D(f )

✈➔


Header Page 13 of 128.

∀x ∈ M ⇒ a±1 x ∈ M,
f (ax) = f (x), ∀x ∈ M.

❱➼ ❞ö ✶✳✸✳

f (x) = sin(2πlog2 x)✳
+
❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý ✷ tr➯♥ R ✳
+
±1
+
❚❤➟t ✈➟②✱ t❛ ❝â ∀x ∈ R t❤➻ 2 x ∈ R ✈➔
❳➨t ❤➔♠ sè

❑❤✐ ✤â

f (x)

❧➔ ❤➔♠ t✉➛♥

f (2x) = sin(2πlog2 (2x))
= sin(2π(1 + log2 x))

❱➼ ❞ö ✶✳✹✳

= sin(2πlog2 x) = f (x), ∀x ∈ R+ .
❈❤♦ ✈➼ ❞ö ✈➲ ❤➔♠ sè ❧✐➯♥ tö❝ ✈➔ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉

❦ý ❝ì sð ✺✿

f (5x) = f (x), ∀x > 0.
●✐↔✐✳

❚❛ ❝â

∀x ∈ R+ ⇒ 5± x ∈ R+

✈➔

log5 (5x) = 1 + log5 x ⇔ 2πlog5 (5x) = 2π + log5 x.
✣➦t

f (x) = cos[2πlog5 x], ∀x > 0,

s✉② r❛

f (5x) = cos[2πlog5 (5x)]
= cos[2π+2πlog5 x]
= cos[2πlog5 x] = f (x)
❱➟②✱ ❤➔♠ sè
t➼♥❤ ❝❤✉ ❦ý

f (x) = cos(2πlog5 x), ∀x > 0 ❧➔ ♠ët ❤➔♠ sè t✉➛♥ ❤♦➔♥ ♥❤➙♥
+
❝ì sð ✺ tr➯♥ R ✳

✶✳✸✳✷ ▲î♣ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳
f (x)
❍➔♠ sè

t➼♥❤ ❝❤✉ ❦ý

a (a ∈
/ {0, 1, −1})

✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥

tr➯♥

M

♥➳✉

M ⊂ D(f )

∀x ∈ M ⇒ a±1 x ∈ M,
f (ax) = −f (x), ∀x ∈ M.
✶✶

Footer Page 13 of 128.

✈➔


Header Page 14 of 128.

❱➼ ❞ö ✶✳✺✳

❈❤♦ ✈➼ ❞ö ❤➔♠ sè ❧✐➯♥ tö❝ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉

❦ý ❝ì sð ✸✿

f (3x) = −f (x), ∀x > 0.
●✐↔✐✳

❚❛ ❝â

∀x ∈ R+ ⇒ 3± x ∈ R+

✈➔

log3 (3x) = 1 + log3 x ⇔ πlog3 (3x) = π + πlog3 x
✣➦t

f (x) = cos[πlog3 x],∀x > 0

s✉② r❛

f (3x) = cos[πlog3 (3x)]
= cos[π + πlog3 x]
= −cos[πlog3 x] = −f(x).

f (x) = cos(πlog3 x),
+
❝ì sð ✸ tr➯♥ R ✳

❱➟②✱ ❤➔♠ sè
t➼♥❤ ❝❤✉ ❦ý

❧➔ ♠ët ❤➔♠ sè ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥

✶✷

Footer Page 14 of 128.


Header Page 15 of 128.

❈❤÷ì♥❣ ✷
P❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✈➔
❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t
✷✳✶ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t

✷✳✶✳✶ P❤÷ì♥❣ ♣❤→♣ ♠ô ❤â❛ ✈➔ ✤÷❛ ✈➲ ❝ò♥❣ ❝ì sè
✶✳ P❤÷ì♥❣ ♣❤→♣ ❝❤✉♥❣✳
✣➸ ❝❤✉②➸♥ ➞♥ sè ❦❤ä✐ ❧♦❣❛r✐t t❛ ❝â t❤➸ ♠ô ❤â❛ t❤❡♦ ❝ò♥❣ ♠ët ❝ì sè ❝↔
❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ ❈❤ó♥❣ t❛ ❧÷✉ þ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝ì ❜↔♥ s❛✉✿

• loga f (x) = b ⇔

0.

f (x) = ab

• loga f (x) = loga g(x) ⇔

0f (x) = g(x) .

✷✳ ❈→❝ ✈➼ ❞ö✳
❱➼ ❞ö ✷✳✶✳

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

logx (x2 + 4x − 4) = 3

✭✷✳✶✮

●✐↔✐✳
✣✐➲✉ ❦✐➺♥

√
x > −2 + √8
x + 4x − 4 > 0 ⇔
0 x < −2 − 8
02




✶✸

Footer Page 15 of 128.

⇔

√

8 − 2 < x = 1.

✭✷✳✷✮


Header Page 16 of 128.

❚❛ sû ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐✿

3 = logx x3

❑❤✐ ✤â

(2.1) ⇔ logx (x2 + 4x − 4) = logx x3 ⇔ x2 + 4x − 4 = x3
x=1
3
2
2
⇔ x − x − 4x + 4 = 0 ⇔ (x − 1)(x − 4) = 0 ⇔ x = 2
x = −2
x = −2 ❦❤æ♥❣
❧➔ x = 1, x = 2✳

❚r♦♥❣ ✤â
♥❣❤✐➺♠

❱➼ ❞ö ✷✳✷✳

t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✷✳✷✮✳ ❱➟②✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â

❳→❝ ✤à♥❤ ♠ ✤➸ ♣❤÷ì♥❣ tr➻♥❤

2log4 (2x2 − x + 2m − 4m2 ) + log 1 (x2 + mx − 2m2 ) = 0

✭✷✳✸✮

2
❝â ♥❣❤✐➺♠

x1 , x2

t❤ä❛ ♠➣♥

x21 + x22 > 1✳

●✐↔✐✳
❚❛ sû ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐✿

1
log4 (2x2 − x + 2m − 4m2 ) = log2 (2x2 − x + 2m − 4m2 )
2
log 1 (x2 + mx − 2m2 ) = −log2 (x2 + mx − 2m2 ).

2
❑❤✐ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ❝â ❞↕♥❣

log2 (2x2 − x + 2m − 4m2 ) − log2 (x2 + mx − 2m2 ) = 0
⇔ log2 (2x2 − x + 2m − 4m2 ) = log2 (x2 + mx − 2m2 )
x2 + mx − 2m2 > 0
⇔
2
2x2 − x + 2m − 4m2 = x2 + mx − 2m
 2
 x + mx − 2m2 > 0
x2 + mx − 2m2 > 0
⇔
⇔
.
x = 2m
x2 − (m + 1)x + 2m − 2m2 = 0
 x1 = 1 − m
2
✣➸ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ❝â ♥❣❤✐➺♠
❧➔✿

x1 , x2


 (2m)2 + m(2m) − 2m2 > 0
(1 − m)2 + m(1 − m) − 2m2 > 0 ⇔

(2m)2 + (1 − m)2 > 1

✶✹

Footer Page 16 of 128.

t❤ä❛ ♠➣♥

x21 + x22 > 1

✤✐➲✉ ❦✐➺♥

−1 < m < 0
2
1 .
5
2


Header Page 17 of 128.

✷✳✶✳✷ P❤÷ì♥❣ ♣❤→♣ ✤➦t ➞♥ ♣❤ö
▼ö❝ ✤➼❝❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❧➔ ❝❤✉②➸♥ ❝→❝ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ ✈➲
♣❤÷ì♥❣ tr➻♥❤ ❤♦➦❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè q✉❡♥ ❜✐➳t✳

✷✳✶✳✷✳✶ ❉ò♥❣ ➞♥ ♣❤ö ❝❤✉②➸♥ ♣❤÷ì♥❣ tr➻♥❤ ❧♦❣❛r✐t t❤➔♥❤ ♠ët
♣❤÷ì♥❣ tr➻♥❤ ✈î✐ ♠ët ➞♥ ♣❤ö
✶✳ P❤÷ì♥❣ ♣❤→♣ ❝❤✉♥❣
❚❛ ❧÷✉ þ ❝→❝ ♣❤➨♣ ✤➦t ➞♥ ♣❤ö t❤÷í♥❣ ❣➦♣ s❛✉✿

❉↕♥❣ ✶✿

◆➳✉ ✤➦t

t = loga x

✈î✐

x>0

t❤➻

logka x = tk ,
1
logx a = ✈î✐ 0 < x = 1.
t
❉↕♥❣ ✷✿ ❚❛ ❜✐➳t r➡♥❣ alogb c = clogb a ✱ ❞♦ ✤â ♥➳✉ ✤➦t t = alogb x t❤➻ t = xlogb a ✳
log x
❚✉② ♥❤✐➯♥ tr♦♥❣ ♥❤✐➲✉ ❜➔✐ t♦→♥ ❝â ❝❤ù❛ a b ✱ t❛ t❤÷í♥❣ ✤➦t ➞♥ ♣❤ö ❞➛♥
✈î✐ t = logb x✳

✷✳ ❈→❝ ✈➼ ❞ö
❱➼ ❞ö ✷✳✸✳

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

(x − 2)log3 [9(x−2)] = 9(x − 2)3 .

✭✷✳✹✮

●✐↔✐✳

✣✐➲✉ ❦✐➺♥

x − 2 > 0 ⇔ x > 2.

▲➜② ❧♦❣❛r✐t ❝ì sè ✸ ❤❛✐ ✈➳ ♣❤÷ì♥❣ tr➻♥❤

✭✷✳✹✮✱ t❛ ✤÷ñ❝✿

log3 [(x − 2)log3 [9(x−2)] ] = log3 [9(x − 2)3 ]
⇔ [log3 [9(x − 2)]].log3 (x − 2) = 2 + log3 (x − 2)3
⇔ [2 + log3 (x − 2)].log3 (x − 2) = 2 + 3log3 (x − 2).
✣➦t

t = log3 (x − 2).

❑❤✐ ✤â ✭✷✳✺✮ ❝â ❞↕♥❣✿

(2 + t)t = 2 + 3t ⇔ t2 − t − 2 = 0 ⇔
•

❱î✐

t = −1

•

❱î✐

t=2

t❛ ❝â

t❛ ❝â

7
log3 (x − 2) = −1 ⇔ x = .
3

log3 (x − 2) = 2 ⇔ x = 11.
✶✺

Footer Page 17 of 128.

t = −1
t=2 .

✭✷✳✺✮


Header Page 18 of 128.

❱➟②✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠

❱➼ ❞ö ✷✳✹

x=

7
3

✈➔

x = 11.

✳

✭✣➲ t❤✐ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ❑❤è✐ ❉ ♥➠♠ ✶✾✾✽✮ ●✐↔✐ ♣❤÷ì♥❣

tr➻♥❤

1
loga (ax).logx (ax) = loga2 ( ), 0 < a = 1.
a

✭✷✳✻✮

●✐↔✐✳
✣✐➲✉ ❦✐➺♥✿

ax > 0
0 < x = 1 ⇔ 0 < x = 1.

❇✐➳♥ ✤ê✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮ ✈➲ ❞↕♥❣

1
(loga a + loga x).(logx a + logx x) = − loga a
2
1
1
+ 1) = − .

⇔ (1 + loga x).(
loga x
2
✣➦t

t = loga x✳

✭✷✳✼✮

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✼✮ ❝â ❞↕♥❣

1
1
(1 + t).( + 1) = − ⇔ 2t2 + 5t + 2 = 0 ⇔
t
2
•

❱î✐

t=−

1
2

t❛ ❝â

1
1
loga x = − ⇔ x = √ .

2
a

•

❱î✐

t = −2

t❛ ❝â

loga x = −2 ⇔ x =

❱➟②✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠

1
x=√
a

✈➔

1
t=− .
2
t = −2

1
.
a2
x=

1
.
a2

✷✳✶✳✷✳✷ ❉ò♥❣ ➞♥ ♣❤ö ❝❤✉②➸♥ ♣❤÷ì♥❣ tr➻♥❤ ❧♦❣❛r✐t t❤➔♥❤ ♠ët
♣❤÷ì♥❣ tr➻♥❤ ✈î✐ ♠ët ➞♥ ♣❤ö ♥❤÷♥❣ ❤➺ sè ✈➝♥ ❝❤ù❛ ①
✶✳ P❤÷ì♥❣ ♣❤→♣ ❝❤✉♥❣✳
❚❛ ❧÷✉ þ ❝â ♥❤ú♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❦❤✐ ❧ü❛ ❝❤å♥ ➞♥ ♣❤ö ❝❤♦ ♠ët ❜✐➸✉
t❤ù❝ t❤➻ ❝→❝ ❜✐➸✉ t❤ù❝ ❝á♥ ❧↕✐ ❦❤æ♥❣ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ tr✐➺t ✤➸ q✉❛ ➞♥ ♣❤ö
✤â ❤♦➦❝ ♥➳✉ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ t❤➻ ❝æ♥❣ t❤ù❝ ❜✐➸✉ ❞✐➵♥ ❧↕✐ q✉→ ♣❤ù❝ t↕♣✳
❑❤✐ ✤â t❛ ❝â t❤➸ ✤➸ ♣❤÷ì♥❣ tr➻♥❤ ð ❞↕♥❣✿ ✧❝❤ù❛ ➞♥ ♣❤ö ♥❤÷♥❣ ❤➺ sè ✈➝♥
❝❤ù❛ ①✧✳

✶✻

Footer Page 18 of 128.


Header Page 19 of 128.

❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② t❛ t❤÷í♥❣ ✤÷ñ❝ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ t❤❡♦
➞♥ ♣❤ö ✭❤♦➦❝ ✈➝♥ t❤❡♦ ➞♥ ①✮ ❝â ❜✐➺t sè

✷✳ ❈→❝ ✈➼ ❞ö
❱➼ ❞ö ✷✳✺✳

∆

❧➔ ♠ët sè ❝❤➼♥❤ ♣❤÷ì♥❣✳

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

lg2 x − lg x.log2 (4x) + 2log2 x = 0.

✭✷✳✽✮

●✐↔✐✳
✣✐➲✉ ❦✐➺♥

x > 0✳

❇✐➳♥ ✤ê✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮ ✈➲ ❞↕♥❣

lg2 x − (2 + log2 x) lg x + 2log2 x = 0.
✣➦t

t = lg x✱

❦❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ t÷ì♥❣ ✤÷ì♥❣ ✈î✐

t2 − (2 + log2 x).t + 2log2 x = 0
∆ = (2 + log2 x)2 − 8log2 x = (2 − log2 x)2 .
❙✉② r❛

t=2
t = log2 x .

•

❱î✐

t=2

•

❱î✐

t = log2 x

t❛ ❝â

lg x = 2 ⇔ x = 100.

t❛ ❝â

❱➟②✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠

❱➼ ❞ö ✷✳✻✳

lg x
⇔ lg x = 0 ⇔ x = 1.
lg 2
x = 100, x = 1.

lg x = log2 x ⇔ lg x =

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

lg2 (x2 + 1) + (x2 − 5). lg(x2 + 1) − 5x2 = 0.

✭✷✳✾✮

●✐↔✐✳
✣➦t

t = lg(x2 +1)✱ ✤✐➲✉ ❦✐➺♥ t ≥ 0 ✈➻ x2 +1 ≥ 1 ♥➯♥ lg(x2 +1) ≥ lg 1 = 0.

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✾✮ t÷ì♥❣ ✤÷ì♥❣ ✈î✐

t2 + (x2 − 5).t − 5x2 = 0
∆ = (x2 − 5)2 + 20x2 = (x2 + 5)2 .
❙✉② r❛

t=5
t = −x2 .
✶✼

Footer Page 19 of 128.


Header Page 20 of 128.

•

❱î✐

t=5

•

❱î✐

t = −x2

t❛ ❝â

√
lg(x2 + 1) = 5 ⇔ x2 + 1 = 105 ⇔ x = ± 99999.

t❛ ❝â

lg(x2 + 1) = −x2 ⇔

❱➟②✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❜❛ ♥❣❤✐➺♠

lg(x2 + 1) = 0
⇔ x = 0.
x2 = 0

√
x = ± 99999

✈➔

x = 0.

✷✳✶✳✷✳✸ ❉ò♥❣ ➞♥ ♣❤ö ❝❤✉②➸♥ ♣❤÷ì♥❣ tr➻♥❤ ❧♦❣❛r✐t t❤➔♥❤ ♠ët
❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈î✐ ✷ ➞♥ ♣❤ö
✶✳ P❤÷ì♥❣ ♣❤→♣ ❝❤✉♥❣

❇➡♥❣ ✈✐➺❝ sû ❞ö♥❣ tø ❤❛✐ ➞♥ ♣❤ö trð ❧➯♥ ✭❣✐↔ sû ❧➔ ✉✱ ✈✮✱ t❛ ❝â t❤➸
❦❤➨♦ ❧➨♦ ✤÷❛ ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➲ ✈✐➺❝ ①➨t ♠ët ❤➺✱ tr♦♥❣ ✤â✿

•

P❤÷ì♥❣ tr➻♥❤ t❤ù ♥❤➜t ❝â ✤÷ñ❝ tø ♣❤÷ì♥❣ tr➻♥❤ ✤➛✉ ❜➔✐✳

•

P❤÷ì♥❣ tr➻♥❤ t❤ù ❤❛✐ ❝â ✤÷ñ❝ tø ✈✐➺❝ ✤→♥❤ ❣✐→ ♠é✐ q✉❛♥ ❤➺ ❝õ❛ ✉✱
✈✳

✷✳ ❈→❝ ✈➼ ❞ö
❱➼ ❞ö ✷✳✼✳

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

3 + log2 (x2 − 4x + 5) + 2 5 − log2 (x2 − 4x + 5) = 6.

✭✷✳✶✵✮

●✐↔✐✳
✣✐➲✉ ❦✐➺♥


 x2 − 4x + 5 > 0
3 + log2 (x2 − 4x + 5) ≥ 0 ⇔ x2 − 4x + 5 ≤ 25 ⇔ x2 − 4x − 27 ≤ 0

2
5 − log

√ 2 (x − 4x + 5)
ó 0
⇔ 2 − 29 ≤ x ≤ 2 + 29.
✣➦t

u=
v=

3 + log2 (x2 − 4x + 5)
,
5 − log2 (x2 − 4x + 5)

✤✐➲✉ ❦✐➺♥

✶✽

Footer Page 20 of 128.

u, v ≥ 0✳


Header Page 21 of 128.

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✵✮ ✤÷ñ❝ ❝❤✉②➸♥ t❤➔♥❤

u = 6 − 2v
⇔
2
2
(6

−
2v)
+
v
=
8

u=2
 v=2
u = 6 − 2v

v=2

2
.
u=
14 ⇔ 

5
v=

14
5

v=
5

u + 2v = 6
u2 + v 2 = 8 ⇔

⇔







•

u = 6 − 2v
5v 2 − 24v + 28 = 0

❱î✐

3 + log2 (x2 − 4x + 5) = 2
u=2
⇔
v=2
5 − log2 (x2 − 4x + 5) = 2
⇔ log2 (x2 − 4x + 5) = 1 ⇔ x2 − 4x + 5 = 2
x=1
⇔ x2 − 4x + 3 = 0 ⇔ x = 3 .
•

❱î✐

3 + log2 (x2 − 4x + 5) = 52
u = 52
⇔

v = 14
5 − log2 (x2 − 4x + 5) = 14
5
5
⇔ log2 (x2 − 4x + 5) = − 71
25
71
2
− 71
25
⇔ x − 4x + 5 = 2
⇔ x2 − 4x + 5 − 2− 25 = 0.✭✈æ

♥❣❤✐➺♠✮

❱➙②✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t✳

❱➼ ❞ö ✷✳✽✳

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

log2 (x −

x2 − 1) + 3log2 (x +

x2 − 1) = 2.

●✐↔✐✳
✣✐➲✉

✣➦t

 2
 x −√1 ≥ 0
❦✐➺♥
x − √x2 − 1 > 0 ⇔ x ≥ 1.

2−1 >0
x + x√
u = log2 (x − √ x2 − 1)
.
v = log2 (x + x2 − 1)

◆❤➟♥ ①➨t r➡♥❣

u + v = log2 (x −
= log2 (x −

x2 − 1) + log2 (x +
x2 − 1).(x +

= log2 1 = 0.
✶✾

Footer Page 21 of 128.

x2 − 1)

x2 − 1)

Header Page 22 of 128.

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✤÷ñ❝ ❝❤✉②➸♥ t❤➔♥❤

u+v =0
u = −v
u = −1
u + 3v = 2 ⇔ 2v = 2 ⇔ v = 1
√
√
x − √x2 − 1 = 12
log2 (x − √x2 − 1) = −1
⇔
⇔
⇔ x = 45 .
2
log2 (x + x2 − 1) = 1
x+ x −1=2
❱➟②✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠

5
x= .
4

✷✳✶✳✷✳✹ ❉ò♥❣ ➞♥ ♣❤ö ❝❤✉②➸♥ ♣❤÷ì♥❣ tr➻♥❤ ❧♦❣❛r✐t t❤➔♥❤ ♠ët
❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈î✐ ✶ ➞♥ ♣❤ö ✈➔ ♠ët ➞♥ ①
✶✳ P❤÷ì♥❣ ♣❤→♣ ❝❤✉♥❣
❇➯♥ ❝↕♥❤ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤➦t ➞♥ ♣❤ö tr➯♥✱ t❛ ❝â t❤➸ sû ❞ö♥❣ ♣❤÷ì♥❣
♣❤→♣ ✧❝❤✉②➸♥ ♣❤÷ì♥❣ tr➻♥❤ t❤➔♥❤ ❤➺ ❣ç♠ ❤❛✐ ➞♥ ❧➔ ♠ët ➞♥ ♣❤ö ✈➔ ➞♥

①✧ ❜➡♥❣ ❝→❝❤ t❤ü❝ ❤✐➺♥ t❤❡♦ ❝→❝ ❜÷î❝✿

❇÷î❝ ✶✿

❇✐➳♥ ✤ê✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➲ ❞↕♥❣✿

❇÷î❝ ✷✿

✣➦t

u = ϕ(x)✱

f [x, ϕ(x)] = 0.

t❛ ❜✐➳♥ ✤ê✐ ♣❤÷ì♥❣ tr➻♥❤ t❤➔♥❤ ❤➺✿

u = ϕ(x)
f (x, u) = 0 .

✷✳ ❈→❝ ✈➼ ❞ö
❱➼ ❞ö ✷✳✾✳

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

log22 x +

log2 x + 1 = 1.

●✐↔✐✳
✣➦t

u = log2 x.

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✶✮ ✤÷ñ❝ ❝❤✉②➸♥ t❤➔♥❤

u2 +

√

u+1=1

✣✐➲✉ ❦✐➺♥

u+1≥0
1 − u2 ≥ 0 ⇔ −1 ≤ u ≤ 1.
√
√
✣➦t v =
u + 1✱ ✤✐➲✉ ❦✐➺♥ 0 ≤ v ≤ 2
✷✵

Footer Page 22 of 128.

s✉② r❛

v 2 = u + 1.

✭✷✳✶✶✮

Header Page 23 of 128.

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✤÷ñ❝ ❝❤✉②➸♥ t❤➔♥❤ ❤➺

u2 = 1 − v
⇒ u2 − v 2 = −(u + v)
v2 = u + 1
u+v =0
⇔ (u + v)(u − v + 1) = 0 ⇔ u − v + 1 = 0 .
•

❱î✐

v = −u✱

t❛ ✤÷ñ❝

√
1− 5
u=
2
2√ .
u −u−1=0⇔

1+ 5
u=
2
√
1+ 5
❚r♦♥❣ ✤â u =

❦❤æ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ −1 ≤ u ≤ 1✳
2
√
√
√
1− 5
1− 5
1− 5
u=
⇔ log2 x =
⇔x=2 2 .
2
2


•

❱î✐

u − v + 1 = 0✱
2

t❛ ✤÷ñ❝

u +u=0⇔

u=0
u = −1 ⇔

❱➟②✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❧➔

❱➼ ❞ö ✷✳✶✵✳

log2 x = 0
log2 x = −1 ⇔

√
1− 5
x = 2 2 ,x = 1

✈➔

x=1
1 .
x=
2

x=

1
2

✳

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

7x−1 = 6log7 (6x − 5) + 1.
●✐↔✐✳

5

6x − 5 > 0 ⇔ x > .
6
y − 1 = log7 (6x − 5).

✣✐➲✉ ❦✐➺♥
✣➦t

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✤÷ñ❝ ❝❤✉②➸♥ t❤➔♥❤ ❤➺

7x−1 = 6(y − 1) + 1
⇔
y − 1 = log7 (6x − 5)
✷✶

Footer Page 23 of 128.

7x−1 = 6y − 5
.
7y−1 = 6x − 5

✭✷✳✶✷✮


Header Page 24 of 128.

rứ t ữỡ tr ừ t ữủ

7x1 + 6x = 7y1 + 6y.
t số

f (t) = 7t1 + 6t



ỡ tr

f (x) = f (y) x = y.
y1
õ ữỡ tr 7
= 6x 5 õ 7x1 6x + 5 = 0.
x1
t số g(x) = 7
6x + 5

õ ữủ t ữợ









5
D = ( ; +).
6

g (x) = 7x1 . ln 7 6
g (x) = 7x1 .ln2 7 > 0, x D

r g (x) ỗ tr

t ỵ ữỡ tr



g(x) = 0

õ ổ q

tr
t r

g(1) = g(2) = 0

ữỡ tr õ

ú ỵ

x=1



x = 2

ố ợ ữỡ tr rt õ ởt t õ

sax+b = clogs (dx + e) + x +
ợ

d = ac +



e = bc +

ợ ữỡ tr t tỹ ữ s

0dx + e > 0 .
ay + b = logs (dx + e).


t

ứ õ tỹ ữợ tữỡ tỹ ữ ử

Pữỡ số t
Pữỡ
ị tữ ừ ừ ữỡ ỗ
ỷ ử số ử s õ t t số



Footer Page 24 of 128.


Header Page 25 of 128.

❜✳ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❝❤ù❛ t❤❛♠ sè ♠✱ t❛ ❝â t❤➸ ❝♦✐ ♠ ❧➔ ➞♥✱ ❝á♥ ① ❧➔

t❤❛♠ sè✱ s❛✉ ✤â t➻♠ ❧↕✐ ① t❤❡♦ ♠✳

✷✳ ❈→❝ ✈➼ ❞ö
❱➼ ❞ö ✷✳✶✶✳

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

lg4 x + lg3 x − 2lg2 x − 9 lg x − 9 = 0.
●✐↔✐✳

x > 0✳
t = lg x✱ t❛ ✤÷ñ❝

✣✐➲✉ ❦✐➺♥
✣➦t

t4 + t3 − 2t2 − 9t − 9 = 0 ⇔ 32 + 3t.3 − t4 − t3 + 2t2 = 0.
✣➦t

u = 3✱

t❛ ✤÷ñ❝

u2 + 3t.u − t4 − t3 + 2t2 = 0.
❚❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ t❤❡♦ ✉✱ ✤÷ñ❝

∆ = 9t2 + 4(t4 + t3 − 2t2 ) = (2t2 + t)2 ✱ s✉② r❛

−3t − (2t2 + t)
3 = −t2 − 2t

u = −t2 − 2t
u=
2
⇔
⇔

3 = t2 − t
u = t2 − t
−3t + (2t2 + t)
u=
2
√
1
±
13
⇔ t2 − t − 3 = 0 ⇔ t =
.
2
√
√
√
1 + 13
1 + 13
1 + 13
2
• ❱î✐ t =
t❛ ✤÷ñ❝ lg x =
⇔ x = 10
.
2

2
√
√
√
1 − 13
1 − 13
1 − 13
2
• ❱î✐ t =
t❛�♥ tö❝ tr➯♥ R+

t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥

x
f ( ) = f (x) − f (y), ∀x, y ∈ R+ .
y

✭✸✳✸✮

❳→❝ ✤à♥❤ ❝→❝ ❤➔♠

●✐↔✐✳
✣➦t

x
= t✳
y

❑❤✐ ✤â

x = ty

✈➔

(3.3) ⇔ f (t) = f (ty) − f (y)
⇔ f (ty) = f (t) + f (y), ∀t, y ∈ R+ .
❚❤❡♦ ❦➳t q✉↔ ❝õ❛ ❱➼ ❞ö ✸✳✶✱ t❤➻

f (x) = b ln x, ∀x ∈ R+ , b ∈ R

tò② þ✳

❑➳t ❧✉➟♥✿

f (x) = b ln x, ∀x ∈ R+ , b ∈ R

❱➼ ❞ö ✸✳✸✳

tò② þ✳

❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè

f :R→R

t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥

f (5x) = f (x), ∀x ∈ R

✭✸✳✹✮

●✐↔✐✳
●✐↔ sû ❢ ❧➔ ❤➔♠ sè t❤ä❛ ♠➣♥ ✤➲ ❜➔✐✱ ❦❤✐ ✤â t❛ ❝â ✭✸✳✹✮✳

•

x > 0✱ ✤➦t x = 5u ✭❤❛② u = log5 x✮✳ ❚❤❛② ✈➔♦ ✭✸✳✹✮ t❛ ✤÷ñ❝
f (5u+1 ) = f (5u ), ∀u ∈ R.
u
✣➦t f (5 ) = g(u), ∀u ∈ R✳ ❑❤✐ ✤â g(u + 1) = g(u), ∀u ∈ R. ◆❤÷ ✈➟②
❣ ❧➔ ❤➔♠ sè t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ✶ tr➯♥ R✱ ❝❤➥♥❣ ❤↕♥ g(x) = cos(2πx),
∀x ∈ R. ❚❛ ❝â
❑❤✐

f (x) = f (5u ) = g(u) = g(log5 x), ∀x ∈ (0; +∞).
❚❤û ❧↕✐✿ ❱î✐ ♠å✐

x ∈ (0; +∞)✱

❦❤✐ ✤â

f (5x) = g(log5 (5x)) = g(1 + log5 x) = g(log5 x) = f (x).
❱➟②✱ ❦❤✐
✶ tr➯♥

x>0

t❤➻

f (x) = g(log5 x)

R✮✳
✺✼

Footer Page 59 of 128.

✭❣ ❧➔ ❤➔♠ sè t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý