Đẳng thức và bất đẳng thức trong lớp hàm logarit

✣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◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿

●❙✳❚❙❑❍✳ ◆❣✉②➵♥ ❱➠♥ ▼➟✉

❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✵


✐

▼ö❝ ❧ö❝
▼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✉❛♥ ✤➳♥ ❧î♣ ❤➔♠ ❧ç✐ ✈➔ ❤➔♠ ❧ç✐ ❧♦❣❛r✐t ✳

✽
✾

❈❤÷ì♥❣ ✷✳ ✣➥♥❣ t❤ù❝ ✈➔ ♣❤÷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 ❞ö♥❣ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✻

❈❤÷ì♥❣ ✸✳ ❇➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❧î♣ ❤➔♠ ❧♦❣❛r✐t
✸✳✶

✸✳✷

✸✽

❈→❝ ❞↕♥❣ t♦→♥ ÷î❝ ❧÷ñ♥❣ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❧♦❣❛r✐t

✳ ✳ ✳ ✳ ✳

✸✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

✸✳✶✳✶

❇➜t ✤➥♥❣ t❤ù❝ ❤➔♠ ❧♦❣❛r✐t

✸✳✶✳✷

P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛ ❧♦❣❛r✐t

✳ ✳ ✳ ✳

✹✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✶

✸✳✷✳✶

❇➔✐ t♦→♥ ❝ü❝ trà ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳

✺✶

✸✳✷✳✷

❇➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❞➣② sè ✈➔ ❣✐î✐ ❤↕♥

✺✻

✸✳✷✳✸

Ù♥❣ ❞ö♥❣ ❤➔♠ ❧ç✐✱ ❤➔♠ ❧♦❣❛r✐t tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤

▼ët sè t➼♥❤ t♦→♥ ❦❤→❝ ❧✐➯♥ q✉❛♥

✳ ✳ ✳ ✳ ✳ ✳ ✳

❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑➳t ❧✉➟♥

✻✵

✻✻





t tự õ tr t q trồ tr t ồ ởt ở
q trồ ừ t số tự t tự tr
ợ rt ởt tr ỳ ở ỡ q trồ ừ
ữỡ tr t tr ồ ờ tổ tr ữỡ
tr ỗ ữù ợ P ử ử ý t qố

ỹ
t tr t ồ s ọ t t
q tợ t t ừ rt tữớ ữủ ỳ
t tữớ ữủ tở õ ỏ ọ tữ
s õ ổ õ sự t út sỹ t tỏ
õ s t ừ ồ s
ự ỗ ữù ỗ ữù ồ s ọ
rt tổ ồ t tự t
tự tr ợ rt
t st ởt số ợ t tứ t ố
t t tr ữợ ỳ
trú ỗ ữỡ t
ữỡ ởt số tự q rt r ữỡ
t tr ởt số t t ỡ ừ rt trữ
ừ t t ởt số q ợ ỗ
ỗ rt
ữỡ r tự rt tr ợ số ờ
ữủ tr tổ q ởt số t sỷ ử ữỡ tr
ữỡ tr rt ố ữỡ
tr ữỡ ữỡ tr s t rt
ũ ợ ử tữỡ ự
ữỡ t tự tr ợ rt ữỡ tr
t tự rt ữỡ t tự ự
rt tổ q ử ử t r ỏ tr ự ử
ừ t ỹ tr rt ụ ữ



t t ợ ự ử ỗ rt tr ự ởt
ợ t tự

ữủ t t rữớ ồ ồ ồ
ữợ sỹ ữợ ừ sữ s ồ
tọ ỏ t ỡ s s ố ợ ữớ
t t ữợ tr t tự ự
t tr sốt q tr ồ t t ụ
tọ ỏ t ỡ t tợ ổ tr
trữớ ồ ồ ồ ú ù
t t tr sốt tớ ồ t t rữớ
ỗ tớ tổ ụ ỷ ớ ỡ tợ ỗ
ổ ú ù ở tổ tr q tr t


t



ồ


✸

❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥
q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t
▼ö❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠
❧♦❣❛r✐t❀ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✈➔ ♠ët sè ✤à♥❤ ❧➼ ❧✐➯♥
q✉❛♥ ✤➳♥ ❧î♣ ❤➔♠ ❧ç✐ ✈➔ ❤➔♠ ❧ç✐ ❧♦❣❛r✐t✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣
✤÷ñ❝ 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

✤÷ñ❝

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♥❣ ❤ñ♣ ✷✿

❤ì♥ ♥ú❛

❚❛ ❦❤↔♦ s→t t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè

f (x) = (loga x) =
❱➟②✱ ❦❤✐

I = R✳

x > 0✱

tr♦♥❣ ✷ tr÷í♥❣ ❤ñ♣✳

✲ ❚r÷í♥❣ ❤ñ♣ ✶✿
❑❤✐ ✤â✱

✈➔ t➟♣ ❣✐→ trà

❧✐➯♥ tö❝ ✈➔ ❝â ✤↕♦ ❤➔♠ ✈î✐ ♠å✐

f (x) =

❚➼♥❤ ❝❤➜t ✶✳✶

0 < a = 1✳

1
> 0, ∀x > 0.
x ln a

❧➔ ❤➔♠ ✤ç♥❣ ❜✐➳♥ tr➯♥ ❉✳


✹
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔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✱

✭❚➼♥❤ ❧ç✐✱ ❧ã♠✮ ❳➨t ❤➔♠ sè

t❛ ❝â

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✉② r❛ ❤➔♠ sè ❧ç✐ tr➯♥ (0; +∞)✳

❚➼♥❤ ❝❤➜t ✶✳✸✳

❱î✐ ♠å✐

a > 0, a = 1

✈➔

❚➼♥❤ ❝❤➜t ✶✳✹✳

❱î✐ ♠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 ✶✳✺✳

❱î✐ ♠å✐

x > 0✳

❱î✐ ♠å✐

0 < a = 1, b = 1

✈➔

x > 0✱

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

❚➼♥❤ ❝❤➜t ✶✳✼✳

❍➔♠ sè

❚➼♥❤ ❝❤➜t ✶✳✽✳

❱î✐ ♠å✐

α

❜➜t ❦ý✱ t❛ ❝â

1
loga xα = α logaα x = logaα xα .
α

loga b. logb c = loga c, loga b =

❚➼♥❤ ❝❤➜t ✶✳✻✳

❱î✐

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❛ t❤÷í♥❣ ❧➔♠ q✉❡♥ ✈î✐ ❧î♣ ❤➔♠ ❧÷ñ♥❣ ❣✐→❝
❧➔ ♥❤ú♥❣ ❤➔♠ t✉➛♥ ❤♦➔♥ ✭❝ë♥❣ t➼♥❤✮ q✉❡♥ t❤✉ë❝✳ ❘➜t ♥❤✐➲✉ ♣❤÷ì♥❣ tr➻♥❤
❤➔♠ ✈➔ ❝→❝ ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥ ✤á✐ ❤ä✐ ❝➛♥ t➻♠ ❤✐➸✉ t❤➯♠ ❝→❝ t➼♥❤ ❝❤➜t
✈➔ ✤➦❝ tr÷♥❣ ❝õ❛ ❧î♣ ❤➔♠ t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❣➢♥ ✈î✐
❤➔♠ ❧♦❣❛r✐t✳

✶✳✷✳✶ ❍➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳
❦ý

a; (a > 1)

❍➔♠ sè

M

tr➯♥

♥➳✉

f (x) ✤÷ñ❝ ❣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✳

◆❣÷ñ❝ ❧↕✐✱ ♥➳✉

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)

◆❣÷ñ❝
✭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) ✤÷ñ❝ ❣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➯♥

❈❤ù♥❣ ♠✐♥❤✳

M

✤➲✉ ❧➔ ❤➔♠

M✳

❚❤❡♦ ❣✐↔ t❤✐➳t tç♥ 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
◆❣÷ñ❝ ❧↕✐✱ 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 ❧✉➟♥✿
tò② þ ❝❤✉

❇➔✐ t♦→♥ ✶✳✷✳

❈❤♦

tr♦♥❣ ✤â

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

◆❣÷ñ❝ ❧↕✐✱ ✈î✐ ♠é✐

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
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 ở t tũ ỵ ý tr

R

ởt số q ợ ỗ ỗ rt
ỵ
số f : [a; b] R tọ tử tr


[a; b] õ tr (a; b) f (a) = f (b) t tỗ t c (a; b)
s f (c) = 0

ỵ

r

số f : [a; b] R tọ tử

tr [a; b] tr (a; b) õ tỗ t c (a; b) s

f (c) =

ỵ

f (b) f (a)
.
ba

sỷ x > 1 õ

t tự r

(1 + x) 1 + x 0 1
(1 + x) 1 + x 0 1.


[a, b]

ợ ồ

f : [a, b] R ồ ỗ
x, y [a, b] ồ [0, 1] t õ
số tỹ

tr

f (x + (1 )y) f (x) + (1 )f (y).



tr t õ t tự t t t õ t
õ
ỗ

f

ỗ tỹ sỹ

f

t õ õ ó

f






ỵ

f

: (a, b) R ỗ
tr (a, b) n N 1 , 2 , . . . , n (0, 1) số tỹ tọ
1 + 2 + ã ã ã + n = 1. õ ợ ồ x1 , x2 , . . . , xn (a, b) t õ
t tự s

n

n

f

i xi

i f (xi ),

i=1

i=1



f (1 x1 + 2 x2 + ã ã ã + n xn ) 1 f (x1 ) + 2 f (x2 ) + ã ã ã + n f (xn ).


ỵ



t tự rt ố ợ rt

số {xk , yk I(a, b), k = 1, 2, . . . , n} ợ 0 < a < b tọ

x1 x2 ã ã ã xn , y1 y2 ã ã ã yn

x1 y1 ,






x1 + x2 y1 + y2 ,


......





x1 + x2 + ã ã ã + xn1 y1 + y2 + ã ã ã + yn1 ,




x + x + ããã + x = y + y + ããã + y .
1
2
n
1
2
n
õ ự ợ f (x) = logd x ợ d > 1 t õ

f (x1 ) + f (x2 ) + ã ã ã + f (xn ) f (y1 ) + f (y2 ) + ã ã ã + f (yn ).




ợ số ữỡ

r ở

a, b

r số ồ

A(a, b) =


r

a+b
.
2

r ồ


G(a, b) =



t

ab.

r ỏ

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

.

a1
k

A(a), G(a), H(a) tữỡ ự ữủ ồ tr ở
tr tr ỏ ừ số a1 , a2 , . . . , an .
ữủ



ợ số ổ

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

ữủ ồ tr ụ tứ ừ số

ỵ ợ số ữỡ a

a1 , a2 , ..., an .

= {ak }nk=1 , r1 < r2 t Mr1 (a) <
Mr2 (a). tự r a1 = a2 = ... = an .


ừ số

a, b

ợ số ữỡ

a, b

t

tr rt

tự

L(a, b) =

ba
, 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 số ữỡ a < b õ t tự
H(a, b) < G(a, b) < L(a, b) < A(a, b).
ự

0 < a < b f (x) = ex
[ln a, ln b]. õ 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

õ


✶✷
❚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.

✣à♥❤ ❧þ ✶✳✽✳ ❱î✐ 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♥❣ ♠✐♥❤✳

❱➼ ❞ö ✶✳✹✳
▲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 ✈➲ ✤➥♥❣ t❤ù❝ ❧♦❣❛r✐t tr♦♥❣
❧î♣ ❤➔♠ sè ❝❤✉②➸♥ ✤ê✐ ❝→❝ ✤↕✐ ❧÷ñ♥❣ tr✉♥❣ ❜➻♥❤❀ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②
❞↕♥❣ ❧♦❣❛r✐t ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t ❝ò♥❣
♠ët sè ✈➼ ❞ö✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ✤÷ñ❝ 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) ✈➔ ✈î✐ x = y t❤➻
✳ ●✐↔ sû

f (2x) = 2f (x),
●✐↔ sû ✈î✐

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 ✤â✱ t❤❡♦ ♥❣✉②➯♥ ❧þ q✉② ♥↕♣✱ t❛ ❝â

f (nx) = nf (x),

∀x ∈ N.

✭✷✳✷✮
❑❤✐ ✤â


✶✺
❑➳t ❤ñ♣ ✈î✐ t➼♥❤ ❝❤➜t

f (−x) = −f (x)✱

t❛ ✤÷ñ❝

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 ❤ñ♣ ✭✷✳✸✮ ✈➔ ✭✷✳✹✮✱ t❛ ✤÷ñ❝

f

m
m
=
f (1),
2n
2n

❙û ❞ö♥❣ ❣✐↔ 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❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮✳

❱➟②

◆❤➟♥ ①➨t ✷✳✶✳

tö❝ tr➯♥

R

f (x) = ax,

✈î✐

a∈R

tò② þ.

❚r♦♥❣ ❜➔✐ t♦→♥ tr➯♥✱ ♥➳✉ t❛ t❤❛② ❣✐↔ t❤✐➳t ❤➔♠ sè

❜ð✐ ❤➔♠ sè

❚❤➟t ✈➟②✱ ♥➳✉ ❤➔♠ sè

f

f

x0 ∈ R t❤➻
✤✐➸♠ x0 t❤➻

❧✐➯♥ tö❝ t↕✐

❧✐➯♥ tö❝ t↕✐

f

❧✐➯♥

❦➳t q✉↔ tr➯♥ ✈➝♥ ✤ó♥❣✳

lim f (x) = f (x0 )

x→x0
✈➔ ✈î✐ ♠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.

✭✷✳✺✮

ớ



ứ s r

f (x) x x R

t

x0 + y
2

f (x0 )f (y) = f
tự

tỗ t

x0



f (x0 ) = 0

, x, y R

f (x) 0.

t trữớ ủ

f (x) > 0 x R
x+y
2

ln f

=

õ tữỡ ữỡ ợ

ln f (x) + ln f (y)
, x, y R
2



x+y
2

g

=

g(x) + g(y)
, x, y R
2

g(x) = ln f (x) ữỡ tr t g(x) = ax+b

tr õ

r ừ t õ

f (x) = eax+b , a, b R
ỷ t t

tũ ỵ.

f (x) = eax+b , a, b R tũ

ỵ tọ

ừ t t r t

t
ớ

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



sỷ số

f

tử tr

R+

tọ

1

f ( xy) = (f (x) + f (y)), x > 0, y > 0.
2
t
s

x = eu , y = ev g(u) = f (eu ) õ g(u) tử
r ợ ồ u, v R t õ

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

ứ

ữỡ tr t

g(u) = au + b

tr õ

a, b



số

f (eu ) = au + b.
t

x = eu u = ln x

t õ

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

ừ t t r

t

số


f ( xy) =

f

tử tr

R+

tọ

f (x)f (y), x, y R+ .

ớ





ứ ừ t s r
s

f (x0 ) = 0

f (x) 0, x R+

tỗ t

x0 > 0

t tứ s r


f ( x0 y) =

f (x0 )f (y) = 0, y R+ .

f (x) = 0 f (x) > 0 ợ ồ x R+
x = eu , y = ev , f (eu ) = g(u)
õ g(u) tử tr R õ

g u + v = g(u)g(v), u, v R.

r trữớ ủ

t t t

t q ừ t t õ

g(u) 0
ỷ t t



g(u) = eau+b a, b R

tũ ỵ.

f (x) 0 f (x) = ea ln x+b = cxa c > 0 tọ

ừ t t r

f (x) 0

t



f (x) = ea ln x+b = cxa c > 0.

số

f

tử tr

R+

tọ

2f (x)f (y)

f ( xy) =
, x, y R+ .
f (x) + f (y)

ớ



sỷ

f

tử tọ


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❝ ✈î✐ ♠å✐

❧➔ ❤➔♠ ❧✐➯♥ tö❝ ❦❤✐

a > 0✳

x>0

♠➔

✈î✐ ♠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
◆❣÷ñ❝ ❧↕✐ ♥➳✉ 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

❦❤æ♥❣ ❧✐➯♥ tö❝ 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✐ ✈î✐ ❤➔♠ ❧♦❣❛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 ❞↕♥❣

❛✮ ❚r÷î❝ ❤➳t t❛ t➻♠ ❤➔♠ sè

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♦♥❣ ✭✷✳✾✮ ❧➜②

y=x

✈➔ ❞♦ ✤â

tò② þ.

(−∞, 0)✱

♠✉è♥ ✈➟②✱ ①➨t

✈➔ sû ❞ö♥❣ ❦➳t q✉↔ ♣❤➛♥ ❛✮

t❛ ✤÷ñ❝

1
1
f (x) = f (x2 ) = b ln x2 = b ln |x|,
2
2
❚❤û ❧↕✐ t❛ t❤➜② ❤➔♠

f (x) = b ln |x|

✈î✐

∀x ∈ R− ,
b∈R

✈î✐

b∈R

tò② þ.

tò② þ✱ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉

❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ ✤➦t r❛✳ ❱➟②

f (x) = b ln |x|,

❇➔✐ t♦→♥ ✷✳✼✳

∀x ∈ R \ {0},

❳→❝ ✤à♥❤ ❝→❝ ❤➔♠

f (x)

✈î✐

b∈R

❧✐➯♥ tö❝ tr➯♥

tò② þ.

R+

t❤ä❛ ♠➣♥ ✤✐➲✉

❦✐➺♥

x
y

f

▲í✐ ❣✐↔✐
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❤➜② ❤➔♠

f (x) = b ln x,

tò② þ.

∀x ∈ R+ , b ∈ R

tò② þ✱ t❤ä❛ ♠➣♥ ❝→❝

✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ ✤➦t r❛✳ ❱➟②

∀x ∈ R+ , b ∈ R

f (x) = b ln x,

❇➔✐ t♦→♥ ✷✳✽✳

❳→❝ ✤à♥❤ ❤➔♠ sè

f : R → R✱

tò② þ.

❧✐➯♥ tö❝ tr➯♥

R

✈➔ t❤ä❛ ♠➣♥

✤✐➲✉ ❦✐➺♥

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

ứ

t õ

f (x)f (2x0 x) = f
r

f (x) = 0, x R
f (x) > 0, x R

rữớ ủ

2

x + 2x0 x
2

= [f (x0 )]2 > 0.

t số

g

ữ s

g : R R, g(x) = ln(f (x)), x R.
õ

g

tử 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 số

h(0) = 0





h ữ s h : R R, h(x) = g(x) g(0), x R
h tử tr R ứ t õ
x+y
2

h
r

y=0

=

h(x) + h(y)
,
2

õ

x, y R.

t ữủ

h

x
1
= h(x),
2
2

x, y R.



ứ s r

x, y R.

h(x + y) = h(x) + h(y),



h(x) = x x R
à = g(0) õ

ứ t ữỡ tr t ữủ
ợ

ởt số tỹ tũ ỵ õ t

g(x) = x+à ln(f (x)) = x+à f (x) = ex+à = aex (a = eà > 0).
ỷ t số

f (x) = aex , x R a,

số

a > 0

tọ

ừ

rữớ ủ f (x) < 0,

x R

õ

f (x) > 0, x R

trữớ ủ s r

f (x) = bex ,

x R (b,

số, b

> 0).







f (x) = cex , x R c,

số ỷ t tọ

t số tọ ừ

f (x) = 0, x R

t



f (x) = cex x R (c,

số tử

f :RR

f 2 (x) = f (x + y)f (x y),

ớ

số).

tọ

x, y R.





P (u, v) t x 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 tọ t sỷ 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, ợ g(x) = ln
.

g(x) = 2g
2
f (0)

ỷ t

ứ t õ

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



ứ t õ


g(2x) = g(x + y) + g(x y),

ợ ồ số tỹ

u, v

t

u+v
uv
= x,
= y
2
2

g(u + v) = g(u) + g(v),

x, y R.



ứ t õ

u, v R.




✷✷
❚ø ✭✷✳✷✶✮✱ t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② t❛ ✤÷ñ❝

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 tò② þ✱

a

❱➻ t❤➳

∀x ∈ R.

t❤ä❛ ♠➣♥ ✭✷✳✶✼✮✳ ❱➟②✱ t➜t ❝↔

f (x) = αax , ∀x ∈ R✱

✈î✐

α

❧➔

❧➔ ❤➡♥❣ sè ❞÷ì♥❣✳

✷✳✷ P❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t
P❤÷ì♥❣ tr➻♥❤ ❧♦❣❛r✐t ❝ì ❜↔♥ ❝â ❞↕♥❣✿

loga x = m.

❱î✐ ♠é✐ ❣✐→ trà tò② þ

m✱ ♣❤÷ì♥❣ tr➻♥❤ loga x = m ❧✉æ♥ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❧➔ x = am ✳
m
◆â✐ ❝→❝❤ ❦❤→❝✱ ∀m ∈ (−∞; +∞), loga x = m ⇔ x = a .
❝õ❛

❚r♦♥❣ ♣❤➛♥ ♥➔②✱ tæ✐ tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧♦❣❛r✐t ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤ ✤↕✐ sè✳

P❤÷ì♥❣ ♣❤→♣ ✤÷❛ ✈➲ ❝ò♥❣ ❝ì sè ✈➔ ♠ô ❤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❤✉ ✤÷ñ❝ ♣❤÷ì♥❣
tr➻♥❤ ♠ô✳
❙❛✉ ✤➙② ❧➔ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛✳

❱➼ ❞ö ✷✳✶✳

●✐↔✐ ♣❤÷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

✭✷✳✷✸✮