✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
◆●❯❨➍◆ ◆●➴❈ ◗❯❨➌◆
✣➃◆● ❚❍Ù❈ ❱⑨ ❇❻❚ ✣➃◆● ❚❍Ù❈
❚❘❖◆● ▲❰P ❍⑨▼ ▲❖●❆❘■❚
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✵
✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
◆●❯❨➍◆ ◆●➴❈ ◗❯❨➌◆
✣➃◆● ❚❍Ù❈ ❱⑨ ❇❻❚ ✣➃◆● ❚❍Ù❈
❚❘❖◆● ▲❰P ❍⑨▼ ▲❖●❆❘■❚
❈❤✉②➯♥ ♥❣➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P
▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✸
▲❯❾◆ ❱❿◆
ữớ ữợ ồ
▼➟✉
❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✵
✐
▼ư❝ ❧ư❝
▼Ð ✣❺❯
❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t
✶
✸
✶✳✶
▼ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸
✶✳✷
✣➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✺
✶✳✷✳✶
❍➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✺
✶✳✷✳✷
❍➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
✻
✶✳✷✳✸
❈→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
✶✳✸
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼ët sè ✤à♥❤ ❧➼ ❧✐➯♥ q✉❛♥ ✤➳♥ ❧ỵ♣ ❤➔♠ ỗ ỗ rt
ữỡ tự ♣❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t ✶✹
✷✳✶
P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❞↕♥❣ ❧♦❣❛r✐t
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✹
✷✳✷
P❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✷
✷✳✸
❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✹
✷✳✸✳✶
P❤➨♣ ❝❤✉②➸♥ ✈➲ ❤➺ ✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✹
✷✳✸✳✷
❙û ❞ư♥❣ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ữỡ t tự tr ợ rt
t ữợ ữủ t tự rt
✳ ✳
✸✽
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✽
✸✳✶✳✶
❇➜t ✤➥♥❣ t❤ù❝ ❤➔♠ ❧♦❣❛r✐t
✸✳✶✳✷
P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛ ❧♦❣❛r✐t
✳ ✳ ✳ ✳
✹✹
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✺✶
✸✳✷✳✶
❇➔✐ t♦→♥ ❝ü❝ trà ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳
✺✶
✸✳✷✳✷
❇➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ số ợ
ử ỗ rt tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤
▼ët sè t➼♥❤ t♦→♥ ❦❤→❝ ❧✐➯♥ q✉❛♥
✳ ✳ ✳ ✳ ✳ ✳ ✳
❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❑➳t ❧✉➟♥
✻✵
✻✻
✶
▼ð ✤➛✉
❇➜t ✤➥♥❣ t❤ù❝ ❝â ✈à tr➼ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ tr♦♥❣ t♦→♥ ❤å❝ ✈➔ ❧➔ ♠ët ❜ë
♣❤➟♥ q✉❛♥ trå♥❣ ❝õ❛ ❣✐↔✐ t➼❝❤ ✈➔ ✤↕✐ sè✳ ✣➥♥❣ t❤ù❝✱ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣
❧ỵ♣ ❤➔♠ ❧♦❣❛r✐t ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝ì ❜↔♥ ✈➔ q✉❛♥ trå♥❣ ❝õ❛
❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❜➟❝ tr✉♥❣ ồ ờ tổ tr ữỡ
tr ỗ ữù ❍❙● ð ❝→❝ ❧ỵ♣ ❚❍P❚ ♣❤ư❝ ✈ư ❝→❝ ❦ý t❤✐ ❍❙● q✉è❝ ❣✐❛
✈➔ ❦❤✉ ✈ü❝✳
✣➦❝ ❜✐➺t✱ tr♦♥❣ ❝→❝ ❦➻ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ t♦→♥ ❝→❝ ❝➜♣✱ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥
q✉❛♥ tợ t t ừ rt tữớ ữủ ✤➲ ❝➟♣✳ ◆❤ú♥❣
❞↕♥❣ t♦→♥ ♥➔② t❤÷í♥❣ ✤÷đ❝ ①❡♠ ❧➔ t❤✉ë❝ ❧♦↕✐ ❦❤â ✈➔ ✤á✐ ❤ä✐ t÷ ❞✉②✱ ❦❤↔
♥➠♥❣ ♣❤→♥ ✤♦→♥ ❝❛♦✱ s♦♥❣ ♥â ❧↕✐ ❧✉æ♥ ❝â sù❝ ❤➜♣ ❞➝♥✱ t❤✉ ❤ót sü t➻♠ tá✐✱
â❝ s→♥❣ t↕♦ ❝õ❛ ❤å❝ s✐♥❤✳
✣➸ ✤→♣ ự ỗ ữù ỗ ữù ❤å❝ s✐♥❤ ❣✐ä✐ ✈➲
❝❤✉②➯♥ ✤➲ ❤➔♠ ❧♦❣❛r✐t✱ tæ✐ ❝❤å♥ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✧✣➥♥❣ t❤ù❝ ✈➔ ❜➜t ✤➥♥❣
t❤ù❝ tr♦♥❣ ❧ỵ♣ ❤➔♠ ❧♦❣❛r✐t✧✳
❚✐➳♣ t❤❡♦✱ ❦❤↔♦ s→t ♠ët sè ❧ỵ♣ ❜➔✐ t♦→♥ tø ❝→❝ ✤➲ t❤✐ ❍❙● ◗✉è❝ ❣✐❛ ✈➔
❝→❝ t➾♥❤ t❤➔♥❤ tr ữợ ỳ
trú ỗ ữỡ t
ữỡ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t✳ ❚r♦♥❣ ❝❤÷ì♥❣
♥➔② t→❝ ❣✐↔ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧♦❣❛r✐t✱ ✤➦❝ tr÷♥❣
❝õ❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✈➔ ♠ët sè ✤à♥❤ ❧➼ ❧✐➯♥ q✉❛♥ ✤➳♥ ❧ỵ♣ ỗ
ỗ rt
ữỡ r tự rt tr ợ số ờ
ữủ tr✉♥❣ ❜➻♥❤ t❤ỉ♥❣ q✉❛ ♠ët sè ❜➔✐ t♦→♥✱ sû ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤
❤➔♠ ❈❛✉❝❤② ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❞↕♥❣ ❧♦❣❛r✐t✳ ❈✉è✐ ❝❤÷ì♥❣
❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ữỡ tr s t rt
ũ ợ ử tữỡ ự
ữỡ t tự tr ợ rt ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔②
✈➲ ❜➜t ✤➥♥❣ t❤ù❝ ❤➔♠ ❧♦❣❛r✐t ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛
❧♦❣❛r✐t t❤ỉ♥❣ q✉❛ ❝→❝ ✈➼ ❞ö ❝ö t❤➸✳ ◆❣♦➔✐ r❛ ❝á♥ tr➻♥❤ ❜➔② ❝→❝ ù♥❣ ❞ö♥❣
❝õ❛ ❝→❝ ✤à♥❤ ❧➼ ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❝ü❝ trà ❤➔♠ ❧♦❣❛r✐t ❝ơ♥❣ ♥❤÷ ❝→❝ ❜➔✐
t t ợ ự ử ỗ ❧♦❣❛r✐t tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ♠ët
❧ỵ♣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❦✐♥❤ ✤✐➸♥✳
▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ồ ồ
ữợ sỹ ữợ ừ ●✐→♦ s÷✱ ❚✐➳♥ s➽ ❦❤♦❛ ❤å❝ ◆❣✉②➵♥
❱➠♥ ▼➟✉✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ ữớ
t t ữợ tr t t❤ù❝✱ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦
t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ơ♥❣
①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❝→❝ ❚❤➛② ❈ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥✲❚✐♥
tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ ❣✐↔♥❣ ❞↕②✱ ❣✐ó♣ ✤ï ✈➔
t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t rữớ
ỗ tớ tổ ụ ỷ ớ ỡ tợ ỗ
ổ ú ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥
✈➠♥✳
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✸ ♥➠♠ ✷✵✷✵✳
❚→❝ ❣✐↔
◆❣✉②➵♥ ◆❣å❝ ◗✉②➳♥
✸
❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥
q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t
▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠
❧♦❣❛r✐t❀ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✈➔ ♠ët sè ✤à♥❤ ❧➼ ❧✐➯♥
q✉❛♥ ✤➳♥ ợ ỗ ỗ rt t q ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣
✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✳
✶✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧♦❣❛r✐t
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ a > 0, a = 1
f (x) = loga x
❈❤♦
❤➔♠ sè ❧♦❣❛r✐t
✳ ❑❤✐ ✤â ❤➔♠ sè
✤÷đ❝
a✳
log x
x
❚ø ✤à♥❤ ♥❣❤➽❛ ♥➔② t❛ s✉② r❛✿ loga a = 1✱ loga 1 = 0✱ x = a a ✱ x = loga a ✳
❣å✐ ❧➔
❝ì sè
❚r♦♥❣ ❝→❝ ♣❤➛♥ t✐➳♣ t❤❡♦✱ t❛ ❣✐↔ sû
◆❤➟♥ ①➨t ✶✳✶✳
D = (0; +∞)
✐✮ ❍➔♠ sè ❧♦❣❛r✐t ❝â t➟♣ ①→❝ ✤à♥❤
✐✐✮ ❍➔♠ sè
f (x) = loga x
f (x) = loga x
✳
✭❚➼♥❤ ✤ì♥ ✤✐➺✉✮
ln a > 0
a>1
1
.
x ln a
a > 1✳
♥➯♥ s✉② r❛
f (x) = loga x
0 < a < 1✳
t❤➻
✲ ❚r÷í♥❣ ❤đ♣ ✷✿
❤ì♥ ♥ú❛
❚❛ ❦❤↔♦ s→t t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè
f (x) = (loga x) =
❱➟②✱ ❦❤✐
I = R✳
x > 0✱
tr♦♥❣ ✷ tr÷í♥❣ ❤đ♣✳
✲ ❚r÷í♥❣ ❤đ♣ ✶✿
❑❤✐ ✤â✱
✈➔ t➟♣ ❣✐→ trà
❧✐➯♥ tö❝ ✈➔ ❝â ✤↕♦ ❤➔♠ ✈ỵ✐ ♠å✐
f (x) =
❚➼♥❤ ❝❤➜t ✶✳✶
0 < a = 1✳
1
> 0, x > 0.
x ln a
ỗ tr ❉✳
✹
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②
loga x
f (x) < 0, ∀x ∈ D✳
❱➟②✱ ❦❤✐
0
t❤➻
f (x) =
❧➔ ❤➔♠ sè ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ❉✳
❚➼♥❤ ❝❤➜t ✶✳✷
✳
y = loga x, a > 0, a = 1, x > 0
ỗ ó t số
t õ
f (x) = (loga x) =
f (x) =
✲ ◆➳✉
✲ ◆➳✉
−1
.
x2 ln a
1
,
x ln a
a > 1 tù❝ ln a > 0 t❤➻ y < 0 s✉② r❛ ❤➔♠ sè ❧ã♠ tr➯♥ (0; +∞)✳
0 < a < 1 tù❝ ln a < 0 t y > 0 s r số ỗ tr➯♥ (0; +∞)✳
❚➼♥❤ ❝❤➜t ✶✳✸✳
❱ỵ✐ ♠å✐
a > 0, a = 1
✈➔
❚➼♥❤ ❝❤➜t ✶✳✹✳
❱ỵ✐ ♠å✐
a > 0✱ a = 1
✈➔
x1 , x2 ∈ (0; +∞)✱ t❛ ❝â
x1
loga (x1 x2 ) = loga x1 + loga x2 , loga
= loga x1 − loga x2 .
x2
loga xα = αloga x, loga x =
❚➼♥❤ ❝❤➜t ✶✳✺✳
❱ỵ✐ ♠å✐
x > 0✳
❱ỵ✐ ♠å✐
0 < a = 1, b = 1
✈➔
x > 0✱
0 < a = 1, 0 < c = 1
loga x =
❚➼♥❤ ❝❤➜t ✶✳✼✳
❍➔♠ sè
❚➼♥❤ ❝❤➜t ✶✳✽✳
❱ỵ✐ ♠å✐
α
❜➜t ❦ý✱ t❛ ❝â
1
loga xα = α logaα x = logaα xα .
α
loga b. logb c = loga c, loga b =
❚➼♥❤ ❝❤➜t ✶✳✻✳
❱ỵ✐
t❛ ❝â
1
.
logb a
✈➔
x > 0✱
t❛ ❝â
logc x
.
logc a
f (x) = loga x (0 < a = 1) ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐
1
. ◆➳✉ ❤➔♠ sè u = u(x) ❝â ✤↕♦ ❤➔♠
✤✐➸♠ x ∈ (0; +∞) ✈➔ (loga x) =
x ln a
tr➯♥ ❦❤♦↔♥❣ J ∈ R t❤➻ ❤➔♠ sè y = loga u(x)✱ (0 < a = 1) ❝â ✤↕♦ ❤➔♠ tr➯♥
u (x)
J ✈➔ (loga u(x)) =
.
u(x) ln a
✐✮ ❑❤✐
✐✐✮ ❑❤✐
a>1
t❤➻
a > 0✱ a = 1
✈➔
x1 , x2 ∈ (0; +∞)✱
loga x1 < loga x2 ⇔ x1 < x2 .
0
t❤➻
loga x1 < loga x2 ⇔ x1 > x2 .
t❛ ❝â
✺
✶✳✷ ✣➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
❚r♦♥❣ ❝❤÷ì♥❣ tr t t tữớ q ợ ợ ❧÷đ♥❣ ❣✐→❝
❧➔ ♥❤ú♥❣ ❤➔♠ t✉➛♥ ❤♦➔♥ ✭❝ë♥❣ t➼♥❤✮ q✉❡♥ t❤✉ë❝✳ ❘➜t ♥❤✐➲✉ ♣❤÷ì♥❣ tr➻♥❤
❤➔♠ ✈➔ ❝→❝ ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥ ✤á✐ ❤ä✐ ❝➛♥ t➻♠ ❤✐➸✉ t❤➯♠ ❝→❝ t➼♥❤ ❝❤➜t
✈➔ ✤➦❝ trữ ừ ợ t t ♥❤➙♥ t➼♥❤ ❣➢♥ ✈ỵ✐
❤➔♠ ❧♦❣❛r✐t✳
✶✳✷✳✶ ❍➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳
❦ý
a; (a > 1)
❍➔♠ sè
M
tr➯♥
♥➳✉
f (x) ✤÷đ❝ ❣å✐
M ⊂ D(f ) ✈➔
❧➔
❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
❝❤✉
∀x ∈ M s✉② r❛ a±1 x ∈ M
f (ax) = f (x), ∀x ∈ M.
❱➼ ❞ö ✶✳✶✳
f (x) = sin(2π log2 x)✳ ❑❤✐ ✤â f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥
+
+
±1
+
❦ý ✷ tr➯♥ R ✳ ❚❤➟t ✈➟②✱ t❛ ❝â ∀x ∈ R t❤➻ 2 x ∈ R ✈➔
❳➨t
♥❤➙♥ t➼♥❤ ❝❤✉
f (2x) = sin(2π log2 (2x))
= sin(2π(1 + log2 x))
= sin(2π log2 x) = f (x).
❚➼♥❤ ❝❤➜t ✶✳✾✳
◆➳✉
❦ý t÷ì♥❣ ù♥❣ ❧➔
a
f (x)
✈➔
b
✈➔
tr➯♥
g(x)
M
✈➔
❧➔ ❤❛✐ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉
ln |a|
m
= , m, n ∈ N∗
ln |b|
n
t❤➻
F (x) =
f (x) + g(x) ✈➔ G(x) = f (x).g(x) ❧➔ ❝→❝ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ tr➯♥ M ✳
❈❤ù♥❣ ♠✐♥❤✳
ln |a|
m
n
m
=
s✉② r❛ |a| = |b| ✳
ln |b|
n
❝õ❛ F (x) ✈➔ G(x)✳ ❚❤➟t ✈➟②✱ t❛
❚ø ❣✐↔ t❤✐➳t
T := a2n = b2m
❧➔ ❝❤✉ ❦ý
❚❛ ❝❤ù♥❣ ♠✐♥❤
❝â
F (T x) = f (a2n x) + g(b2m x) = f (x) + g(x) = F (x), ∀x ∈ M ;
G(T x) = f (a2n x)g(b2m x) = f (x)g(x) = G(x), ∀x ∈ M.
∀x ∈ M, T ±1 x ∈ M ✳
t➼♥❤ tr➯♥ M ✳
❍ì♥ ♥ú❛✱
♥❤➙♥
❚➼♥❤ ❝❤➜t ✶✳✶✵✳
tr➯♥
tr➯♥
R t❤➻
R+ ✳
❉♦ ✤â✱
F (x), G(x)
❧➔ ❝→❝ ❤➔♠ t✉➛♥ ❤♦➔♥
f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤ ❝❤✉ ❦ý a✱ a > 0
g(t) = f (ln t)✱ ✭t > 0✮ ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý ea
◆➳✉
✻
f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý a ✭a > 1✮
g(t) = f (et ) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤ ❝❤✉ ❦ý ln a tr➯♥ R✳
◆❣÷đ❝ ❧↕✐✱ ♥➳✉
R+
t❤➻
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f (x) ❧➔ ❤➔♠ t✉➛♥
tr➯♥ R✳ ❳➨t g(t) = f (ln t)✱ ✭t > 0✮✳
❤♦➔♥ ❝ë♥❣ t➼♥❤ ❝❤✉ ❦ý
tr➯♥
a✱ a > 0
❚❛ ❝â
g(ea t) = f (ln(ea t)) = f (ln ea + ln t)
= f (a + ln t) = f (ln t) = g(t), ∀t ∈ R+ .
❱➟②
g(t)
◆❣÷đ❝
✭0
❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý
❧↕✐✱
❣✐↔
sû
f (x)
❧➔
< a = 1✮ tr➯♥ R+ ✳
t
❳➨t g(t) = f (e ), ∀t ∈ R✳
❤➔♠
t✉➛♥
ea
tr➯♥
❤♦➔♥
R+ ✳
♥❤➙♥
t➼♥❤
❝❤✉
❦ý
a
❚❛ ❝â
g(t + ln a) = f (et+ln a ) = f (et .eln a )
= f (aet ) = f (et ) = g(t), ∀t ∈ R.
❱➟②
g(t)
❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤ ❝❤✉ ❦ý
ln a
tr➯♥
R✳
✶✳✷✳✷ ❍➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳
❝❤✉ ❦ý
f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
a (a > 1) tr➯♥ M ♥➳✉ M ⊂ D(f ) ✈➔
∀x ∈ M s✉② r❛ a±1 x ∈ M
f (ax) = −f (x), ∀x ∈ M.
❱➼ ❞ö ✶✳✷✳
❍➔♠ sè
f (x) = cos(π log2 x)✳ ❑❤✐ ✤â f (x) ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥
+
❦ý 2 tr➯♥ R ✳
❳➨t
♥❤➙♥ t➼♥❤ ❝❤✉
❚❤➟t ✈➟②✱ t❛ ❝â
∀x ∈ R+
t❤➻
f (2x) = cos(π log2 (2x)) = cos(π+π log2 x) = − cos(π log2 x) = −f (x), ∀x ∈ R+ .
❱➼ ❞ö ✶✳✸✳
√
1
[sin(2π log2 ( 2x)) − sin(2π log2 x)]✳
2
√
f (x) ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥
t➼♥❤ ❝❤✉ ❦ý
2 tr➯♥ R+ ✳
√
+
±1
+
❚❤➟t ✈➟②✱ t❛ ❝â ∀x ∈ R t❤➻ ( 2) x ∈ R ✈➔
√
√
1
f ( 2x) = [sin(2π log2 (2x)) − sin(2π log2 ( 2x))]
2
❳➨t
f (x) =
❑❤✐ ✤â
✼
√
1
= [sin(2π(1 + log2 x)) − sin(2π log2 ( 2x))]
2
√
1
= [sin(2π log2 x) − sin(2π log2 ( 2x))] = −f (x).
2
❚➼♥❤ ❝❤➜t ✶✳✶✶✳
▼å✐ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ tr➯♥
t✉➛♥ t tr
ự
M
M
tt tỗ t↕✐
b > 1 s❛♦ ❝❤♦ ∀x ∈ M
t❤➻
b±1 ∈ M
✈➔
f (bx) = −f (x), ∀x ∈ M.
❙✉② r❛✱
∀x ∈ M
t❤➻
b±1 ∈ M
✈➔
f (b2 x) = f (b(bx)) = −f (bx) = −(−f (x)) = f (x), ∀x ∈ M.
◆❤÷ ✈➟②✱
f (x)
❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý
❚➼♥❤ ❝❤➜t ✶✳✶✷✳ f (x)
tr➯♥
M
❦❤✐ ✈➔ ❝❤➾ ❦❤✐
b2
tr➯♥
M✳
❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý
f (x)
b ✭b > 1✮
❝â ❞↕♥❣✿
1
f (x) = (g(bx) − g(x)),
2
tr♦♥❣ ✤â✱
g(x)
❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý
b2
tr➯♥
M✳
❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ●✐↔ sû f ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý b tr➯♥
M ✳ ❑❤✐ ✤â g(x) = −f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý b2 tr➯♥ M
✈➔
✭✐✐✮
1
1
(g(bx) − g(x)) = (−f (bx) − (−f (x)))
2
2
1
= (−(−f (x)) + f (x)) = f (x), ∀x ∈ M.
2
1
◆❣÷đ❝ ❧↕✐✱ f (x) = (g(bx) − g(x)), t❤➻
2
1
1
f (bx) = (g(b2 x) − g(bx)) = (g(x) − g(bx))
2
2
1
= − (g(bx) − g(x)) = −f (x), ∀x ∈ M.
2
∀x ∈ M
tr➯♥ M ✳
❍ì♥ ♥ú❛✱
♥❤➙♥ t➼♥❤
t❤➻
b±1 x ∈ M ✳
❉♦ ✤â✱
f (x)
❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥
✽
✶✳✷✳✸ ❈→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤
❇➔✐ t♦→♥ ✶✳✶✳
❈❤♦
a > 1✳
❳→❝ ✤à♥❤ t➜t ❝↔ ❝→❝ ❤➔♠
f (x)
t❤ä❛ ♠➣♥ ✤✐➲✉
❦✐➺♥
f (ax) = f (x), ∀x ∈ R+ .
▲í✐ ❣✐↔✐
✳
✣➦t
x = at
✈➔
f (at ) = h1 (t)✳
❑❤✐ ✤â
t = loga x
✈➔
f (ax) = f (x) ⇔ h1 (t + 1) = h1 (t), ∀t ∈ R,
h(t) = f (at ).
x < 0✳ ✣➦t −x = at
tr♦♥❣ ✤â
❳➨t
✈➔
f (−at ) = h2 (t)✳
❑❤✐ ✤â
t = loga |x|
✈➔
f (ax) = f (x) ⇔ h2 (t + 1) = h2 (t), ∀t ∈ R.
f (x) = h(loga |x|)
❦ý ✶ tr➯♥ R
t
tũ ỵ
t
tr õ
a < 0, a = −1✳
h(t)
❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤
❳→❝ ✤à♥❤ t➜t ❝↔ ❝→❝ ❤➔♠
f (x)
t❤ä❛
♠➣♥ ✤✐➲✉ ❦✐➺♥
f (ax) = −f (x), ∀x ∈ R.
▲í✐ ❣✐↔✐
✳ ❚ø ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ s✉② r❛
f (a2 x) = f (x), ∀x ∈ R.
❱➟②✱ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝â ❞↕♥❣
1
f (x) = [g(x) − g(ax)],
2
tr♦♥❣ ✤â
g(a2 x) = g(x), ∀x ∈ R.
❚❤➟t ✈➟②✱ ♥➳✉
f (x)
❝â ❞↕♥❣ tr➯♥ t❤➻ t❛ ❝â
1
f (ax) = [g(ax) − g(a2 x)]
2
ữủ ợ ộ
1
[g(ax) g(x)] = f (x), ∀x ∈ R.
2
f (x) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜➔✐ t♦→♥✱
❑❤✐ ✤â
g(a2 x) = g(x), ∀x ∈ R.
❝❤å♥
g(x) = f (x)✳
✾
✈➔
1
1
[g(x) − g(ax)] = [f (x) − f (ax)]
2
2
1
= [f (x) + f (x)] = f (x), ∀x ∈ R.
2
❙✉② r❛ ♥❣❤✐➺♠ ❝➛♥ t➻♠ ❧➔
1
f (x) = [g(x) − g(ax)],
2
tr♦♥❣ õ
ợ
1
h
log|a| x x > 0
1
2
g(x) = d tũ ỵ ❦❤✐ x = 0
1
h2 log|a| |x| ❦❤✐ x < 0
2
h1 (t)✱ h2 (t)
❧➔ ❝→❝ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤ tò② þ ❝❤✉ ❦ý ✶ tr➯♥
R✳
✶✳✸ ▼ët sè ✤à♥❤ ❧➼ ❧✐➯♥ q ợ ỗ ỗ rt
ỵ ✶✳✶
✳ ❈❤♦ ❤➔♠ sè f : [a; b] → R t❤ä❛ ♠➣♥ ❢ ❧✐➯♥ tö❝ tr➯♥
✭❘♦❧❧❡✮
[a; b]✱ ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (a; b) ✈➔ f (a) = f (b) t tỗ t c (a; b)
s f (c) = 0
ỵ
r
số f : [a; b] → R t❤ä❛ ♠➣♥ ❢ ❧✐➯♥ tö❝
tr➯♥ ✤♦↕♥ [a; b] tr (a; b) õ tỗ t c (a; b) s
f (c) =
ỵ ✶✳✸
f (b) − f (a)
.
b−a
✳ ●✐↔ sû x > −1✳ ❑❤✐ ✤â
✭❇➜t ✤➥♥❣ t❤ù❝ ❇❡r♥♦✉❧❧✐✮
(1 + x)α ≤ 1 + αx ❦❤✐ 0 ≤ α ≤ 1
(1 + x)α ≥ 1 + αx ❦❤✐ α ≤ 0 ∨ α ≥ 1.
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳
[a, b]
♥➳✉ ✈ỵ✐ ♠å✐
f : [a, b] → R ồ ỗ
x, y [a, b] ♠å✐ λ ∈ [0, 1]✱ t❛ ❝â
❍➔♠ sè t❤ü❝
tr➯♥ ❦❤♦↔♥❣
f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y).
✭✶✳✶✮
◆➳✉ tr♦♥❣ ✭✶✳✶✮ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ t t t õ t
õ
ỗ
f
ỗ tỹ sü✳ ❈❤♦ ❤➔♠
f
t❛ ♥â✐ ♥â ❧➔ ❤➔♠ ❧ã♠ ♥➳✉
−f
❧➔ ❤➔♠
ỵ
f
: (a, b) R ỗ
tr (a, b) n N 1 , λ2 , . . . , λn ∈ (0, 1) ❧➔ ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥
λ1 + λ2 + · · · + λn = 1. ❑❤✐ ✤â ✈ỵ✐ ♠å✐ x1 , x2 , . . . , xn ∈ (a, b) t❛ ❝â
✭❇➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥✮
n
n
f
≤
λi xi
λi f (xi ),
i=1
i=1
♥❣❤➽❛ ❧➔
f (λ1 x1 + λ2 x2 + · · · + λn xn ) ≤ λ1 f (x1 ) + λ2 f (x2 ) + · · · + n f (xn ).
ỵ
t tự ❞↕♥❣ ❑❛r❛♠❛t❛ ✤è✐ ✈ỵ✐ ❤➔♠ ❧♦❣❛r✐t✮
❤❛✐ ❞➣② sè {xk , yk ∈ I(a, b), k = 1, 2, . . . , n} ✈ỵ✐ 0 < a < b✱ t❤ä❛ ♠➣♥ ❝→❝
✤✐➲✉ ❦✐➺♥
x1 ≥ x2 ≥ · · · ≥ xn , y1 ≥ y2 ≥ · · · ≥ yn
✈➔
x1 ≥ y1 ,
x1 + x2 ≥ y1 + y2 ,
......
x1 + x2 + · · · + xn−1 ≥ y1 + y2 + · · · + yn−1 ,
x + x + ··· + x = y + y + ··· + y .
1
2
n
1
2
n
❑❤✐ ✤â✱ ù♥❣ ✈ỵ✐ ❤➔♠ f (x) = logd x ✈ỵ✐ d > 1✱ t❛ ✤➲✉ ❝â
f (x1 ) + f (x2 ) + · · · + f (xn ) ≤ f (y1 ) + f (y2 ) + · · · + f (yn ).
ợ số ữỡ
r ở
a, b
r ❜➻♥❤ sè ❤å❝✮✿
A(a, b) =
✐✐✮
❚r✉♥❣ ❜➻♥❤ ♥❤➙♥
a+b
.
2
✭❚r✉♥❣ ❜➻♥❤ ❤➻♥❤ ❤å❝✮✿
√
G(a, b) =
✐✐✐✮
t❛ ✤à♥❤ ♥❣❤➽❛✿
ab.
❚r✉♥❣ ❜➻♥❤ ✤✐➲✉ ❤á❛ ✿
H(a, b) =
2ab
2
=
.
1 1
a+b
+
a b
✭✶✳✸✮
rữớ ủ ỵ
ợ
n
a = {ak }nk=1
ừ
n
số ổ
số ữỡ ố t ỵ ❤✐➺✉
1
A(a) =
n
n
n
n
1/n
ak , H(a) =
ak , G(a) =
n
k=1
k=1
k=1
.
a−1
k
A(a), G(a), H(a) t÷ì♥❣ ù♥❣ ✤÷đ❝ ❣å✐ ❧➔ tr✉♥❣ ❜➻♥❤ ❝ë♥❣✱
tr✉♥❣ ❜➻♥❤ ♥❤➙♥ ✈➔ tr✉♥❣ ❜➻♥❤ ✤✐➲✉ ❤á❛ ❝õ❛ ❝→❝ sè a1 , a2 , . . . , an .
❈→❝ ✤↕✐ ❧÷đ♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✻✳
❱ỵ✐ ❤❛✐ sè ❦❤ỉ♥❣ ➙♠
a, b
✈➔
ar + b r
Mr (a, b) =
2
ữủ
Mr (a, b)
ữủ ồ
r=0
1
r
t ỵ
.
tr ụ tứ
ừ số
a
b.
ố
ợ trữớ ủ ú t ỵ
a=
ữủ
{ak }nk=1 (ak
Mr (a)
1
0), Mr (a) =
n
n
ark
1
r
, r = 0.
k=1
✤÷đ❝ ❣å✐ ❧➔ tr✉♥❣ ❜➻♥❤ ❧ơ② t❤ø❛ ừ số
ỵ ợ số ữỡ a
a1 , a2 , ..., an .
= {ak }nk=1 , r1 < r2 t❤➻ Mr1 (a) <
Mr2 (a). ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a1 = a2 = ... = an .
✣à♥❤ ♥❣❤➽❛ ✶✳✼✳
❝õ❛ ❝→❝ sè
a, b
❱ỵ✐ ❝→❝ sè ❞÷ì♥❣
a, b
t❛ ✤à♥❤ ♥❣❤➽❛
tr✉♥❣ ❜➻♥❤ ❧♦❣❛r✐t
❧➔ ❜✐➸✉ t❤ù❝
L(a, b) =
b−a
, a = b,
ln b − ln a
L(a, a) = a.
t r
L(a, b) =
1
1
ba
b
a
dx
x
1
=
M1
1
, (a, b)
x
.
ỵ ợ ❝→❝ sè ❞÷ì♥❣ a < b ❝â ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝
H(a, b) < G(a, b) < L(a, b) < A(a, b).
❈❤ù♥❣ ♠✐♥❤✳
0 < a < b✳ ❍➔♠ f (x) = ex ❧➔ ❤➔♠
✤♦↕♥ [ln a, ln b]. ❚❛ ❝â ❜➜t tự
sỷ
ỗ tr
ln a + ln b
2
e
(ln b ln a) <
ln b
ex dx <
ln a
ỗ tr
R,
ln a + ln b
(ln b − ln a).
2
❞♦ ✤â
✶✷
❚ø ✤➙② s✉② r❛
√
ab <
b−a
a+b
<
⇔ G(a, b) < L(a, b) < A(a, b).
ln b − ln a
2
❚❛ ❝â
√
2ab
< ab ⇔ H(a, b) < G(a, b).
a+b
❱➟②
H(a, b) < G(a, b) < L(a, b) < A(a, b), a, b > 0, a = b.
ỵ ợ 0 < a < b ❝â ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝
L(a, b) < M1/3 (a, b), L(a3 , b3 ) < A3 (a, b).
❈❤ù♥❣ ♠✐♥❤✳
d
c
✭✶✳✺✮
❚❤❡♦ ❝æ♥❣ t❤ù❝ ❙✐♠♣s♦♥✱ t❛ ❝â
3 2c + d
3 c + 2d
1
f (x)dx = f (c)(d − c) + f
(d − c) + f
(d − c)
8
8
3
8
3
(c − d)5 (4)
1
f (η),
+ f (c)(d − c) −
8
6480
tr♦♥❣ ✤â
c < η < d.
❈❤♦
c = ln a, d = ln b, f (x) = ex
t❛ ❝â
ln b
ex dx
b−a=
ln a
2 ln a + ln b
ln a + 2 ln b
3
3
eln a + 3e
+ 3e
+ eln b
<
(ln b − ln a)
8
3
a1/3 + b1/3
=
(ln b − ln a).
2
❚ø ✤➙② s✉② r❛ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ♥❤➜t tr♦♥❣ ✭✶✳✺✮✳ ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝
t❤ù ♥❤➜t✱ t❤❛②
a, b
t÷ì♥❣ ù♥❣ ❜ð✐
▼➺♥❤ ✤➲ ✶✳✶✳ ❚❛ ❝â ❤➺ t❤ù❝
a3 , b3
t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐✳
L(x2 , y 2 )
= A(x, y).
L(x, y)
❈❤ù♥❣ ♠✐♥❤✳
❚❛ ❝â
x2 − y 2
L(x , y ) =
ln x2 − ln y 2
2
2
✭✶✳✻✮
✶✸
(x − y)(x + y)
2(ln x − ln y)
x+y
= L(x, y)
2
= L(x, y)A(x, y).
=
❚ø ✤➙② s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
❱➼ ❞ư ✶✳✹✳
▲í✐ ❣✐↔✐
❈❤♦
b > a > 0,
❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝
b+a
b 2 − a2
<2
ln b − ln a
2
2
.
✳
❚❤❡♦ ✤➥♥❣ t❤ù❝ ✭✶✳✻✮ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮✱ t❛ ❝â
L(a2 , b2 ) = L(a, b)A(a, b) < A2 (a, b).
❚ø ✤â s✉② r❛ ❜➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
✶✹
❈❤÷ì♥❣ ✷✳ ✣➥♥❣ t❤ù❝ ✈➔ ♣❤÷ì♥❣
tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t
◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ✈➲ ✤➥♥❣ tự rt tr
ợ số ờ ữủ tr✉♥❣ ❜➻♥❤❀ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②
❞↕♥❣ ❧♦❣❛r✐t ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t ❝ị♥❣
♠ët sè ✈➼ ❞ư✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø t➔✐ ❧✐➺✉
❬✶✱ ✸✱ ✺❪✳
✷✳✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❞↕♥❣ ❧♦❣❛r✐t
❇➔✐ t♦→♥ ✷✳✶
✳
✭P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✮ ❳→❝ ✤à♥❤ ❝→❝ ❤➔♠
❧✐➯♥ tö❝ tr➯♥
R
✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
f (x + y) = f (x) + f (y),
▲í✐ ❣✐↔✐
s✉② r❛
f : R → R✱
∀x, y ∈ R.
✭✷✳✶✮
f ❧➔ ❤➔♠ sè t❤ä❛ ♠➣♥ ✤➲ ❜➔✐✱ ❦❤✐ ✤â t❛ ❝â ✭✷✳✶✮✳ ❚ø ✭✷✳✶✮
f (0) = 0, f (−x) = −f (x) ✈➔ ✈ỵ✐ x = y t❤➻
✳ ●✐↔ sû
f (2x) = 2f (x),
●✐↔ sû ✈ỵ✐
k
♥❣✉②➯♥ ❞÷ì♥❣✱
∀x ∈ R.
f (kx) = kf (x)✱ ∀x ∈ R, ∀k ∈ N✳
f ((k + 1)x) = f (kx + x)
= f (kx) + f (x)
= kf (x) + f (x)
= (k + 1)f (x),
∀x ∈ R, ∀k ∈ N.
❚ø õ t ỵ q t õ
f (nx) = nf (x),
∀x ∈ N.
✭✷✳✷✮
❑❤✐ ✤â
t ủ ợ t t
f (x) = f (x)
t ữủ
f (mx) = f (−m(−x)) = −mf (−x) = mf (x),
∀m ∈ Z, ∀x ∈ R.
✭✷✳✸✮
❚ø ✭✷✳✷✮ t❛ ❝â
x
x
x
= 22 f 2 = . . . = 2n f n .
2
2
2
f (x) = 2f
❚ø ✤â s✉② r❛
1
x
=
f (x),
2n
2n
f
∀x ∈ R, ∀n ∈ N.
✭✷✳✹✮
❑➳t ❤đ♣ ✭✷✳✸✮ ✈➔ ✭✷✳✹✮✱ t❛ ✤÷đ❝
f
m
m
=
f (1),
2n
2n
❙û ❞ư♥❣ ❣✐↔ t❤✐➳t ❧✐➯♥ tö❝ ❝õ❛ ❤➔♠
f (x) = ax
f✱
s✉② r❛
∀x ∈ R, a = f (1).
f (x) = ax,
❚❤û ❧↕✐✱ t❛ t❤➜② ❤➔♠
∀m ∈ Z, ∀n ∈ N+ .
t❤ä❛ ♠➣♥ ữỡ tr
t
tử tr
R
f (x) = ax,
ợ
aR
tũ ỵ.
r ❜➔✐ t♦→♥ tr➯♥✱ ♥➳✉ t❛ t❤❛② ❣✐↔ t❤✐➳t ❤➔♠ sè
❜ð✐ ❤➔♠ sè
❚❤➟t ✈➟②✱ ♥➳✉ ❤➔♠ sè
f
f
x0 ∈ R t❤➻
✤✐➸♠ x0 t❤➻
❧✐➯♥ tư❝ t↕✐
❧✐➯♥ tư❝ t↕✐
f
❧✐➯♥
❦➳t q✉↔ tr➯♥ ✈➝♥ ✤ó♥❣✳
lim f (x) = f (x0 )
x→x0
✈➔ ✈ỵ✐ ♠é✐
x1 ∈ R
t❛ ✤➲✉ ❝â
f (x) = f (x − x1 + x0 ) + f (x1 ) − f (x0 ), ∀x ∈ R.
❚ø ✤â s✉② r❛
lim f (x) = lim f (x − x1 + x0 ) + f (x1 ) − f (x0 )
x→x1
x→x1
= f (x0 ) + f (x1 ) − f (x0 ) = f (x1 ).
◆❤÷ ✈➟②✱ ♥➳✉ ❤➔♠
f
①→❝ ✤à♥❤ tr➯♥
♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② t❤➻
❇➔✐ t♦→♥ ✷✳✷✳
❚➻♠ ❝→❝ ❤➔♠ sè
R✱ ❧✐➯♥ tö❝ t↕✐ ✤✐➸♠ x0 ∈ R
f ❧✐➯♥ tö❝ tr➯♥ R✳
f (x)
①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥
R
✈➔ t❤ä❛
✈➔ t❤ä❛
♠➣♥ ✤✐➲✉ ❦✐➺♥
f (x)f (y) = f
x+y
2
, ∀x, y ∈ R.
✭✷✳✺✮
✶✻
▲í✐ ❣✐↔✐
✳
❚ø ✤✐➲✉ ❦✐➺♥ ✭✷✳✺✮ s✉② r❛
f (x) ≥ x ∀x ∈ R✳
t❤➻
x0 + y
2
f (x0 )f (y) = f
tù❝
tỗ t
x0
f (x0 ) = 0
, x, y R
f (x) ≡ 0.
❳➨t tr÷í♥❣ ❤đ♣
f (x) > 0 ∀x R
x+y
2
ln f
=
õ tữỡ ữỡ ợ
ln f (x) + ln f (y)
, ∀x, y ∈ R
2
❤❛②
x+y
2
g
=
g(x) + g(y)
, ∀x, y ∈ R
2
g(x) = ln f (x)✳ ❚❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② t❤➻ g(x) = ax+b✳
tr♦♥❣ ✤â
❙✉② r❛ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝â ❞↕♥❣
f (x) = eax+b , a, b R
ỷ t t
tũ ỵ.
f (x) = eax+b , a, b R tũ
ỵ tọ ✤✐➲✉ ❦✐➺♥
❝õ❛ ❜➔✐ t♦→♥ ✤➦t r❛✳ ❑➳t ❧✉➟♥✳
❇➔✐ t♦→♥ ✷✳✸✳
▲í✐ ❣✐↔✐
f (x) ≡ 0 f (x) = eax+b , a, b R
số
f
tử tr
R+
tũ ỵ.
tọ ♠➣♥ ✤✐➲✉ ❦✐➺♥
f (x) + f (y)
√
f ( xy) =
, ∀x, y ∈ R+ .
2
✳
●✐↔ sû ❤➔♠ sè
f
❧✐➯♥ tö❝ tr➯♥
R+
✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
1
√
f ( xy) = (f (x) + f (y)), ∀x > 0, y > 0.
2
✣➦t
s✉②
x = eu , y = ev ✈➔ g(u) = f (eu )✳ ❑❤✐ ✤â g(u) ❧✐➯♥ tư❝
r❛ ✈ỵ✐ ♠å✐ u, v ∈ R✱ t❛ ❝â
√
1
f
eu ev = (f (eu ) + f (ev ))
2
u + v
f (eu ) + f (ev )
⇒f e 2 =
2
⇒g
u+v
2
=
g(u) + g(v)
.
2
tr➯♥
✭✷✳✻✮
R✳
❚ø ✭✷✳✻✮
✶✼
❚❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② t❤➻
g(u) = au + b✱
tr♦♥❣ ✤â
a, b
❧➔ ❝→❝
❤➡♥❣ sè✳ ❱➟②
f (eu ) = au + b.
✣➦t
x = eu ⇒ u = ln x
✈➔ t❛ ❝â
f (x) = a ln x + b, a, b ∈ R
ỷ t t
tũ ỵ.
f (x) = a ln x + b, a, b R
tũ ỵ tọ ❝→❝ ✤✐➲✉
❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ ✤➦t r❛✳
❇➔✐ t♦→♥ ✷✳✹✳
❚➻♠ ❝→❝ ❤➔♠ sè
√
f ( xy) =
f
❧✐➯♥ tö❝ tr➯♥
R+
✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
f (x)f (y), ∀x, y ∈ R+ .
▲í✐ ❣✐↔✐
✭✷✳✼✮
✳
❚ø ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ s✉② r❛
s❛♦ ❝❤♦
f (x0 ) = 0
f (x) 0, x R+
tỗ t
x0 > 0
t❤➻ tø ✭✷✳✽✮ s✉② r❛
√
f ( x0 y) =
f (x0 )f (y) = 0, ∀y ∈ R+ .
f (x) = 0✳ ◆➳✉ f (x) > 0 ✈ỵ✐ ♠å✐ x ∈ R+
x = eu , y = ev , f (eu ) = g(u)✳
❑❤✐ ✤â g(u) ❧✐➯♥ tö❝ tr➯♥ R ✈➔ ✭✷✳✽✮ ❝â ❞↕♥❣
√
g u + v = g(u)g(v), ∀u, v ∈ R.
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②
t❤➻ t❛ ✤➦t
❚❤❡♦ ❦➳t q✉↔ ❝õ❛ ❇➔✐ t♦→♥ ✷✳✷✱ t❛ ❝â
g(u) ≡ 0
❚❤û ❧↕✐✱ t❛ t
g(u) = eau+b a, b R
tũ ỵ.
f (x) ≡ 0 ❤♦➦❝ f (x) = ea ln x+b = cxa c > 0 t❤ä❛ ♠➣♥
❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ ✤➦t r❛✳ ❱➟②
f (x) ≡ 0
❇➔✐ t♦→♥ ✷✳✺✳
❤♦➦❝
f (x) = ea ln x+b = cxa c > 0.
❚➻♠ ❝→❝ ❤➔♠ sè
f
❧✐➯♥ tö❝ tr➯♥
R+
✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
2f (x)f (y)
√
f ( xy) =
, ∀x, y ∈ R+ .
f (x) + f (y)
▲í✐ ❣✐↔✐
✳
●✐↔ sû
f
❧➔ ❤➔♠ ❧✐➯♥ tư❝ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
√
f ( xy) =
2
1
1
+
f (x) f (y)
∀x > 0, y > 0.
✭✷✳✽✮
✶✽
f (x) = 0 ∀x > 0✱ ❦➳t ❤ñ♣
1
1
+
1
f (x) f (y)
=
.
√
2
f
xy
❚ø ✤✐➲✉ ❦✐➺♥ tr➯♥ s✉② r❛
✣➦t
g(x)
g(x) =
1
✳
f (x)
❱➻
f
❧✐➯♥ tö❝ ✈ỵ✐ ♠å✐
❧➔ ❤➔♠ ❧✐➯♥ tư❝ ❦❤✐
a > 0✳
x>0
♠➔
✈ỵ✐ ♠å✐
x, y > 0✱
f (x) = 0 ∀x > 0
t❛ ❝â
s✉② r❛
▼➦t ❦❤→❝✱ t❛ ❝â
g(x) + g(y)
√
g ( xy) =
.
2
❚❤❡♦ ❇➔✐ t♦→♥ ✷✳✸✱ t❛ ❝â g(x) = a ln x + b✳ ❑❤✐ ✤â
1
f (x) =
, ∀x > 0.
a ln x + b
1
◆❣÷đ❝ ❧↕✐ ♥➳✉ f (x) =
✱ a, b ❧➔ ❝→❝ ❤➡♥❣ sè✳
a ln x + b
◆➳✉ a = 0✱ ①➨t ♣❤÷ì♥❣ tr➻♥❤
b
b
a ln x + b = 0 ⇔ ln x = − ⇔ x = e− a .
a
a=0
❱➟② ♥➳✉
❉♦ ✤â ✤➸
❱➟②
f (x)
t❤➻
f (x)
b
x = x0 = e− a > 0.
1
a = 0 ⇒ f (x) = ✳
b
❦❤ỉ♥❣ ❧✐➯♥ tư❝ t↕✐
❧✐➯♥ tö❝ ❦❤✐
x>0
t❤➻
f (x) = C ✱ tr♦♥❣ ✤â C = 0 ❧➔ ❤➡♥❣ sè✳ ❚❤û ❧↕✐✱ t❛ t❤➜② ❤➔♠ f (x) = C
t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ ✤➦t r❛✳
✣è✐ ✈ỵ✐ ❤➔♠ ❧♦❣❛r✐t
f (t) = loga t, (0 < a = 1, t > 0)✱
t❛ ❝â ❝→❝ ✤➦❝
tr÷♥❣ s❛✉✿
f (xy) = f (x) + f (y)
✈➔
f
x
y
= f (x) − f (y)✱ ∀x, y ∈ R∗+ ✳
❉♦ ❝â ❝→❝ ✤➦❝ tr÷♥❣ ♥➔②✱ ❤➔♠ sè tr➯♥ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠
t÷ì♥❣ ù♥❣✳
❇➔✐ t♦→♥ ✷✳✻
❤➔♠ sè
f (x)
✭P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❞↕♥❣ ❧♦❣❛r✐t✮
❧✐➯♥ tö❝ tr➯♥
R \ {0}
❳→❝ ✤à♥❤ ❝→❝
t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
f (xy) = f (x) + f (y),
▲í✐ ❣✐↔✐
✳
∀x, y ∈ R \ {0}.
✭✷✳✾✮
✳
f (x) tr➯♥ ❦❤♦↔♥❣ (0, +∞)✱ ♠✉è♥ ✈➟②✱
x = eu , y = ev ✈➔ f (et ) = g(t)✳ ❑❤✐ ✤â ✭✷✳✾✮ ❝â ❞↕♥❣
❛✮ rữợ t t t số
x, y R+
t
g(u + v) = g(u) + g(v),
∀u, v ∈ R.
①➨t
✭✷✳✶✵✮
✶✾
❚❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② t❤➻ ✭✷✳✶✵✮
∀x ∈ R+ , a ∈ R
f (x) = a ln x,
❜✮ ❚✐➳♣ t❤❡♦ t❛ t➻♠ ❤➔♠ sè
−
x, y ∈ R
t❤➻
xy ∈ R
+
⇔ g(t) = bt
f (x)
tr
r
y=x
õ
tũ ỵ.
(, 0)✱
♠✉è♥ ✈➟②✱ ①➨t
✈➔ sû ❞ư♥❣ ❦➳t q✉↔ ♣❤➛♥ ❛✮
t❛ ✤÷đ❝
1
1
f (x) = f (x2 ) = b ln x2 = b ln |x|,
2
2
❚❤û ❧↕✐ t❛ t❤➜② ❤➔♠
f (x) = b ln |x|
ợ
x R ,
bR
ợ
bR
tũ ỵ.
tũ ỵ tọ ✤✐➲✉
❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ ✤➦t r❛✳ ❱➟②
f (x) = b ln |x|,
❇➔✐ t♦→♥ ✷✳✼✳
∀x ∈ R \ {0},
❳→❝ ✤à♥❤ ❝→❝
f (x)
ợ
bR
tử tr
tũ ỵ.
R+
tọ
x
y
f
ớ
x
= f (x) − f (y),
∀x, y ∈ R+ .
✭✷✳✶✶✮
✳
✣➦t
y
= t✳
❑❤✐ ✤â
x = ty
✭✷✳✶✶✮
✈➔
⇔ f (t) = f (ty) − f (y)
⇔ f (ty) = f (t) + f (y),
∀t, y ∈ R+ .
❚❤❡♦ ❦➳t q✉↔ ❝õ❛ ❇➔✐ t♦→♥ ✷✳✻✱ t❤➻
∀x ∈ R+ , b ∈ R
f (x) = b ln x,
❚❤û ❧↕✐ t t
f (x) = b ln x,
tũ ỵ.
x R+ , b R
tũ ỵ tọ
❝õ❛ ❜➔✐ t♦→♥ ✤➦t r❛✳ ❱➟②
∀x ∈ R+ , b ∈ R
f (x) = b ln x,
❇➔✐ t♦→♥ ✷✳✽✳
❳→❝ ✤à♥❤ số
f : R R
tũ ỵ.
tử tr
R
tọ ♠➣♥
✤✐➲✉ ❦✐➺♥
f
▲í✐ ❣✐↔✐
x+y
2
2
= f (x)f (y),
∀x, y ∈ R.
✭✷✳✶✷✮
✳
❉➵ t❤➜② ❤➔♠
f ≡0
t❤ä❛ ♠➣♥ ❝→❝ ②➯✉ ❝➛✉ ❝õ❛ ❜➔✐ t♦→♥✳ ❚✐➳♣ t❤❡♦ t❛ ❣✐↔
sỷ
f
ổ trũ ợ
0
tỗ t
x0 R
s
f (x0 ) = 0✳
❚ø
✭✷✳✶✷✮ t❛ ❝â
f (x)f (2x0 − x) = f
❙✉② r❛
f (x) = 0, ∀x ∈ R✳
f (x) > 0, ∀x ∈ R✳
❚r÷í♥❣ ❤đ♣ ✶✳
2
x + 2x0 − x
2
= [f (x0 )]2 > 0.
❳➨t ❤➔♠ sè
g
♥❤÷ s❛✉✿
g : R → R, g(x) = ln(f (x)), x ∈ R.
❑❤✐ ✤â
g
❧✐➯♥ tö❝ tr➯♥
g
R
x+y
2
✈➔
= ln f
x+y
2
= ln
f (x)f (y)
1
= [ln(f (x)) + ln(f (y))]
2
1
= [g(x) + g(y)], ∀x, y ∈ R.
2
❳➨t ❤➔♠ sè
h(0) = 0
✈➔
✭✷✳✶✸✮
h ♥❤÷ s❛✉✿ h : R → R, h(x) = g(x) − g(0), x ∈ R✳
h ❧✐➯♥ tö❝ tr➯♥ R✳ ❚ø ✭✷✳✶✸✮✱ t❛ ❝â
x+y
2
h
❚r♦♥❣ ✭✷✳✶✹✮ ❧➜②
y=0
=
h(x) + h(y)
,
2
❑❤✐ ✤â✱
∀x, y ∈ R.
✭✷✳✶✹✮
t❛ ✤÷đ❝
h
x
1
= h(x),
2
2
∀x, y ∈ R.
✭✷✳✶✺✮
❚ø ✭✷✳✶✹✮ ✈➔ ✭✷✳✶✺✮ s✉② r❛
∀x, y ∈ R.
h(x + y) = h(x) + h(y),
✭✷✳✶✻✮
h(x) = λx✱ ∀x ∈ R✱
µ = g(0)✳ õ
ứ t ữỡ tr t ữủ
ợ
ởt số tỹ tũ ỵ õ t
g(x) = λx+µ ⇒ ln(f (x)) = λx+µ ⇒ f (x) = eλx+µ = aeλx (a = eµ > 0).
❚❤û ❧↕✐ t❤➜② ❤➔♠ sè
f (x) = aeλx , ∀x ∈ R ✭a, λ
❧➔ ❤➡♥❣ sè✱
a > 0✮
t❤ä❛
♠➣♥ ❝→❝ ②➯✉ ❝➛✉ ❝õ❛ ✤➲ ❜➔✐✳
❚r÷í♥❣ ❤đ♣ ✷✳ f (x) < 0,
∀x ∈ R✳
❑❤✐ ✤â✱
−f (x) > 0, ∀x ∈ R✳
tr÷í♥❣ ❤đ♣ ✶ s✉② r❛
−f (x) = beαx ,
∀x ∈ R (b, α
❧➔ ❤➡♥❣ sè, b
> 0).
❚❤❡♦
✷✶
❱➟②
f (x) = ceβx , ∀x ∈ R ✭c, β
❧➔ ❤➡♥❣ sè✮✳ ❚❤û ❧↕✐ t❤➜② t❤ä❛ ♠➣♥✳
❚➜t ❝↔ ❝→❝ ❤➔♠ sè t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❝õ❛ ✤➲ ❜➔✐ ❧➔
f (x) = 0, ∀x ∈ R
❇➔✐ t♦→♥ ✷✳✾✳
✈➔
f (x) = ceβx ∀x ∈ R (c, β
❳→❝ ✤à♥❤ ❤➔♠ sè ❧✐➯♥ tö❝
f :R→R
f 2 (x) = f (x + y)f (x − y),
▲í✐ ❣✐↔✐
❧➔ ❤➡♥❣ sè).
t❤ä❛ ♠➣♥
∀x, y ∈ R.
✭✷✳✶✼✮
✳
P (u, v) ❝❤➾ ✈✐➺❝ t❤❛② x ❜ð✐ u✱
u ∈ R s❛♦ ❝❤♦ f (u) = 0 õ
ỵ
t
t
y
v
P (x, u x) ⇒ f 2 (x) = 0 ⇒ f (x) = 0,
sỷ tỗ
x R.
f (x) 0 t❤ä❛ ♠➣♥ ✭✷✳✶✼✮✳ ❚✐➳♣ t❤❡♦ ❣✐↔ sû f (x) = 0✱ ∀x ∈ R.
x x
x
P
,
⇒ f2
= f (x)f (0)
2 2
2
2 x
f
f (x)
f (x)
⇒
= 2 2 ✈➔
>0
f (0)
f (0)
f (0)
x
f
f (x)
2 , ∀x ∈ R
⇒ ln
= 2 ln
f (0)
f (0)
x
f (x)
, ∀x ∈ R, ✈ỵ✐ g(x) = ln
.
✭✷✳✶✽✮
⇒ g(x) = 2g
2
f (0)
❚❤û ❧↕✐ t❤➜②
❚ø ✭✷✳✶✼✮✱ t❛ ❝â
f 2 (x) f (x + y) f (x − y)
=
.
, ∀x, y ∈ R
f 2 (0)
f (0)
f (0)
f (x + y)
f (x − y)
f (x)
= ln
+ ln
, ∀x, y ∈ R
⇔2 ln
f (0)
f (0)
f (0)
⇔2g(x) = g(x + y) + g(x − y),
∀x, y ∈ R
✭✷✳✶✾✮
❚ø ✭✷✳✶✽✮ t❛ ❝â
✭✷✳✶✾✮
⇔g(2x) = g(x + y) + g(x − y),
❱ỵ✐ ♠å✐ sè t❤ü❝
u, v ✱
✤➦t
u+v
u−v
= x,
= y✳
2
2
g(u + v) = g(u) + g(v),
∀x, y ∈ R.
✭✷✳✷✵✮
❚ø ✭✷✳✷✵✮ t❛ ❝â
∀u, v ∈ R.
✭✷✳✷✶✮
✷✷
❚ø ✭✷✳✷✶✮✱ t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② t❛ ✤÷đ❝
g(x) ≡ ax✳
f (x)
= eg(x) = eax ⇒ f (x) = f (0)eax ⇒ f (x) = αax ,
f (0)
❚❤û ❧↕✐ t❤➜② ❤➔♠ sè
f (x) = αax , ∀x ∈ R
❝→❝ ❤➔♠ sè t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❝õ❛ ✤➲ ❜➔✐ ❧➔
❤➡♥❣ sè tũ ỵ
a
t
x R.
tọ tt
f (x) = ax , x R
ợ
số ữỡ
P❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t
P❤÷ì♥❣ tr➻♥❤ ❧♦❣❛r✐t ❝ì ❜↔♥ õ
loga x = m.
ợ ộ tr tũ ỵ
m ♣❤÷ì♥❣ tr➻♥❤ loga x = m ❧✉ỉ♥ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❧➔ x = am ✳
m
◆â✐ ❝→❝❤ ❦❤→❝✱ ∀m ∈ (−∞; +∞), loga x = m ⇔ x = a .
❝õ❛
❚r♦♥❣ ♣❤➛♥ ♥➔②✱ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧♦❣❛r✐t ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤ ✤↕✐ sè✳
P❤÷ì♥❣ ♣❤→♣ ✤÷❛ ✈➲ ❝ị♥❣ ❝ì sè ✈➔ ♠ơ ❤â❛
❈→❝❤ ❣✐↔✐
•
✳
✳
0 < a = 1
✰ ❉↕♥❣ ✶✿ P❤÷ì♥❣ tr➻♥❤ loga f (x) = b ⇔
f (x) = ab .
0
✰ ❉↕♥❣ ✷✿ P❤÷ì♥❣ tr➻♥❤ loga f (x) = loga g(x) ⇔
f (x) > 0 ❤♦➦❝ g(x) > 0
f (x) = g(x).
t
t
✣➦t t = loga f (x)✳ ❑❤✐ ✤â a = f (x)✱ b = g(x)✱ tø ✤â t❛ t❤✉ ✤÷đ❝ ♣❤÷ì♥❣
tr➻♥❤ ♠ơ✳
❙❛✉ ✤➙② ❧➔ ♠ët sè ✈➼ ❞ư ♠✐♥❤ ❤å❛✳
❱➼ ❞ư ✷✳✶✳
●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤
lg(x3 + 8) = lg(x + 58) +
▲í✐ ❣✐↔✐✳
1
lg(x2 + 4x + 4).
2
✭✷✳✷✷✮
✣✐➲✉ ❦✐➺♥
3
x +8>0
⇔ x + 2 > 0 ⇔ x > −2.
x + 58 > 0
2
x + 4x + 4 > 0
✭✷✳✷✸✮