专题2.4指数与指数函数六大题型（原卷版）_11.4-2_2026-11-04

      
      
专题2.4 指数与指数函数【六大题型】

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3 (−5)2
3
2 qŸnT    U
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B} 5
C}−5
D}−5
Ï( 1-1T23-24 ~Ç·ÈÉЭ·ÌÍUÑÒÓ(ž™ [T    U
A}
12 (−3)4 =
4 −3
B}
3 (𝑥+ 𝑦)4 = (𝑥+ 𝑦)
3
4
C}
3 −8 = −2
D}
𝑛
𝑚
2
= 𝑛2𝑚
1
2
Ï( 1-2T23-24 ~Ñ·ÔÕ֔·×ØUÙy𝑎+
1
𝑎= 2H𝑎
1
2 + 𝑎−1
2'ÚT    U
A}2
B}4
C} ± 2
D} ± 4
Ï( 1-3T23-24 ~Ç·ÛÜÝÞ·ßàNáU?"(−64)
1
3 + [(−3)4]
1
4−( 2−1)
0 +
3 3 3
8 = T    U
A}−
13
2
B}−
11
2
C}−
1
2
D}
1
2
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Æ 2T23-24 ~Ç·âã”J·Ì­UÚ𝑥 #$4𝑥−2𝑥= 2 5n      
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2
3𝑥−1
≤2 5æn      
}
Ï( 2-2T2024 ~·äå·NáU&'(2𝑥>
1
2
𝑥−𝑥2
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}
Ï( 2-3T23-24 çÇ·ÔÕ·ßàNáUÙy𝑥1¡𝑥2[#$9𝑥−3𝑥+2+3=0 èDH
9𝑥1+9𝑥2
𝑥1+𝑥2 =      
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Æ 3T2024·Õéêë·çìUÙy)
𝑓(𝑥) =
2𝑥
2𝑥−1+1HÑÒí†&ž™ [T    U
A})
𝑓(𝑥)/0îÀ
B})
𝑓(𝑥)¹ºn(0,2)
      
      
C})
𝑓(𝑥) *+Ú(0,1)rï
D})
𝑓(𝑥) *+Ú(1,1)rï
Ï( 3-1T2024·äÉ·ìðñòU)
𝑓(𝑥) = 3𝑥2−2|𝑥| ~ó/0î»®nT    U
A}(−∞,0)
B}(−1,0)
C}(0,1)
D}(1, + ∞)
Ï( 3-2T2024·PQ·ìðñòUÙy)
𝑓(𝑥) =
1
e𝑥+𝑎 *+Ú:(1,𝑓(1))rïH𝑎= T    U
A}1
B}2
C}e
D}e2
Ï( 3-3T2024·ÔÕ·~ìUô)
𝑓(𝑥) = 3−2𝑥2+𝑎𝑥õ»®(1,4)`/0îÂH𝑎 ö¹÷ø[T    U
A}(−∞,4]
B}[4,16]
C}(16, + ∞)
D}[16, + ∞)
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Æ 4T2024·ùÜ·úìUô𝑎= 2𝜋−2,𝑏= 6−1,𝑐= 2
1
3HT    U
A}𝑏> 𝑎> 𝑐
B}𝑐> 𝑎> 𝑏
C}𝑎> 𝑏> 𝑐
D}𝑎> 𝑐> 𝑏
Ï( 4-1T2024·ûë·ìðñòUü𝑎= 0.50.4H𝑏= 0.41.1H𝑐= 1.10.5HT    U
A}𝑎< 𝑐< 𝑏
B}𝑐< 𝑎< 𝑏
C}𝑎< 𝑏< 𝑐
D}𝑏< 𝑎< 𝑐
Ï( 4-2T2023·Çýþ±·~ìUÙy aH𝑏∈RH𝑎> 𝑏HÑÒ&'(­&~š¨ÿ [T    U
A}𝑎+ 2 > 𝑏+ 2 B}2𝑎> 2𝑏
C}𝑎2 > 𝑏2
D}2𝑎> 2𝑏
Ï( 4-3T2024·PQ·úìUüĀ
𝑎H𝑏āĂ1001𝑎+ 1010𝑏= 2023𝑎H1014𝑎+ 1016𝑏= 2024𝑏H𝑎H𝑏
 34ănT    U
A}𝑎> 𝑏
B}𝑎= 𝑏
C}𝑎< 𝑏
D}Ą†12
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Æ 5T2024·PQ·ìðñòUÙy)
𝑓(𝑥) = 3𝑥−2−32−𝑥HāĂ𝑓(𝑥) +𝑓(8−3𝑥) > 0 𝑥 ö¹÷ø[
T    U
A}(−∞,4)
B}(−∞,2)
C}(2, + ∞)
D}(−2,2)
Ï( 5-1T2024·PQ·ìðñòUÙy𝑓(𝑥) = 2𝑥−𝑎+ 1Hą𝑓(𝑥) < 6õ»®(1,2)ƨÿHĀ
𝑎 ö¹
÷ø[T    U
A}(−∞,1]
B}[−1, + ∞)
C}(−1,1]
D}(−1,2]
Ï( 5-2T2024·PQ·ìðñòUÙy)
𝑓(𝑥) =
1
2
|𝑥|
HćĈ𝑓(2𝑎) < 𝑓(𝑎−1)¨ÿ žĀ
𝑎 ö¹
÷ø[T    U
A} 1
3 , + ∞
B} 0, 1
3
C}(0,1)
D}(1, + ∞)
Ï( 5-3T2024·äåĉĊ·~ìUÙy)
𝑓(𝑥) = 2𝑥−3−𝑥H&'(𝑓(𝑥2) < 𝑓(2𝑥+ 3) 5ænT    U
      
      
A}(−1,3)
B}(−∞,−1) ∪(3, + ∞)
C}(−3,1)
D}(−∞,−3) ∪(1, + ∞)
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Æ 6T23-24 ~Ç·ċČčä·ÌÍUÙy)
𝑓(𝑥)[šJõ𝑅Ç Ď)
H–𝑥≥0˜H𝑓(𝑥) = 𝑎⋅2𝑥−
2−𝑥Hą𝑓(−1) =
3
2}
(1)<𝑎 ¹Hď<Đ𝑓(𝑥) 5B(©
(2)ô𝑚𝑓(𝑥)−4𝑥−4−𝑥≤0õ(0, + ∞)ÇƨÿH<𝑚 ö¹÷ø}
Ï( 6-1T23-24 ~Ç·ċČđĒ·ÌÍUÙy)
𝑓(𝑥) = 2𝑥}
(1)–𝑥∈[0,8]˜H&'(𝑓(𝑥+ 1) ≥𝑓(𝑥+ 𝑎)2 ē¨ÿH< a ö¹÷ø©
(2)Ø<)
𝐺(𝑥) = 𝑓(𝑥+ 1) +𝑎𝑓(2𝑥)T𝑎∈RUõ𝑥∈(−∞,0] ¼3¹𝐻(𝑎)}
Ï( 6-2T2024 úÑ·Ĕä·ĕØUü)
𝑓(𝑥) = 𝑎+
𝑏
3𝑥−1(𝑎,𝑏∈𝑅).
(1)ÄÅ)
𝑓(𝑥)õ»®(0, + ∞)¡(−∞,0)Ç /0,T&Ė;ė¶Ę$U©
(2)ô)
𝑓(𝑥)õ´šJº`nę)
H<𝑎%𝑏 ă(©
(3)õT2U ĚěÑH–𝑎= 1˜H&'(𝑓(𝑥) ≥𝑘⋅3−𝑥õ𝑥∈(0, + ∞)ƨÿH<𝑘 ö¹÷ø.
Ï( 6-3T23-24 ~Ç·ċČċĜ·ÌÍUšJõ𝐑Ç ę)
H–𝑥< 0˜H𝑓(𝑥) = 𝑥,−1 < 𝑥< 0
−𝑎−𝑥,𝑥≤−1
 
H´
­𝑎> 0,𝑎≠1Hą𝑓(1) = eH´­e[ĝĞr
 He = 2.71828⋯}
(1)<𝑎 ¹©
      
      
(2)–𝑥≥0˜H<)
𝑓(𝑥) 5B(©
(3)ôğõ𝑥2 > 𝑥1 ≥0HāĂ𝑓(𝑥2) = e𝑓(𝑥1)H<𝑥1 ⋅𝑓(𝑥2) ö¹÷ø}
~9/
1}T2023·PQ·ìðñòU 3 3
9
2+ 3
= T    U
A}
1
3
B} 3
3
C} 3
D}3
2}T2023·ċČĠý·ìðñòUÙy𝑎> 0ą𝑎≠1HÑÒ'(ž™ [T    U
A}𝑎−2 ⋅𝑎3 = 𝑎−6
B}𝑎6
𝑎3 = 𝑎2
C}𝑎6 + 𝑎3 = 𝑎9
D}𝑎−3
2 =
1
𝑎3
3}T2023·ġČ·ìðñòUô𝑎−1−𝑎1 = 4H 𝑎−2 + 𝑎2 ¹nT    U
A}8
B}16
C}2
D}18
4}T2023·ĢģäĤĥĦ·ìðñòUÙy)
𝑓(𝑥) ħA*+Ĩ*ĩĪH𝑓(𝑥) 5B(ī‘nT    U
  
A}𝑓(𝑥) =
e𝑥−e−𝑥
3|𝑥|−2
B}𝑓(𝑥) =
e𝑥−e−𝑥
2−3|𝑥|
C}𝑓(𝑥) =
e𝑥+e−𝑥
3|𝑥|−2
D}𝑓(𝑥) =
2𝑥
|𝑥|−1
5}T2023·ûëĬĭĮ·ìðñòUÙyę)
𝑓(𝑥) = 𝑎𝑥+𝑏⋅𝑎−𝑥(𝑎> 0,𝑎≠1)õ[−1,1]Ç ¼3¹n
8
3H𝑎=
TU
      
      
A}
1
3¬ 3
B}
1
2¬ 2
C}3
D}2
6}T2023·įİ·~ìUÙy𝑎= 0.310.1H𝑏= 0.310.2H𝑐= 0.320.1HT    U
A}𝑎> 𝑏> 𝑐
B}𝑏> 𝑎> 𝑐
C}𝑐> 𝑏> 𝑎
D}𝑐> 𝑎> 𝑏
7}T2023·ÛâıзúìUIJijÑàĴ
“Ùy 2nĄ
Hô( 2)
2n 
HğõĄ
𝑎= 𝑏=
2HćĈ𝑎𝑏n 
©ô( 2)
2nĄ
HöĄ
𝑎= ( 2)
2H𝑏=
2Hĵ˜𝑎𝑏= ( 2)
2
2
= ( 2)
2⋅2
= ( 2)
2 = 2n 
}”oĶÃàĴīkė¶ qķ[T    U
A}( 2)
2[ 
B}( 2)
2[Ą
C}ğõĄ
 aHbHćĈ𝑎𝑏n 
D}rĸ‰Ą
 aHbH½ 𝑎𝑏nĄ
8}T2023·PQ·ìðñòUÙy)
𝑓(𝑥) = 2023𝑎𝑥2+𝑏𝑥+1(𝑎≠0) *+ÚĹĺ𝑥= 2rïHą)
𝑓(𝑥) 
¼4¹n 1H&'(𝑓(𝑥) ≥2023 5ænT    U
A}{𝑥∣0 < 𝑥≤4}
B}{𝑥∣𝑥≥4¬𝑥< 0}
C}{𝑥∣0 ≤𝑥≤4}
D}{𝑥∣𝑥≥4¬𝑥≤0}
ú9Ļ
9}T23-24 ~Ç·ļ⼾Ŀ·ßàNáUÑÒÓ(­~š¨ÿ  T    U
A}
𝑛
𝑚
7
= 𝑛7𝑚
1
7
B}
12 (−3)4 =
3 3
C}
4 𝑥3 + 𝑦4 = (𝑥+ 𝑦)
3
4
D}
3 9 =
3 3
10}T2024·įİÝŀ·ìðñòUÙy)
𝑓(𝑥) =
2𝑥
2𝑥−1+1HÑÒ톞™ [T    U
A})
𝑓(𝑥)/0îÀ
B})
𝑓(𝑥)¹ºn(0,2)
C})
𝑓(𝑥) *+Ú(0,1)rï
D})
𝑓(𝑥) *+Ú(1,1)rï
11}T2024·ÛÜ·ìðñòUÙy)
𝑓(𝑥)[šJºn𝑅 Ď)
H𝑔(𝑥)[šJºn𝑅 ę)
Hą𝑓(𝑥) +𝑔(𝑥)
= 2e𝑥.)
𝐹(𝑥) = 𝑓(2𝑥)−2𝑚𝑓(𝑥)õ[0, + ∞)Ç ¼4¹n−11HÑÒqķž™ [T    U
A}𝑓(𝑥) = e𝑥+ e−𝑥
B}𝑔(𝑥)õĀ
æ𝑅/0îÂ
C}𝑚= 3
D}𝑚= −3.3¬
13
4
ç9Łł
      
      
12}T2024·ÇýŃġ·úìU„ 𝑎2 𝑎T´­𝑎> 0Un 

 e(n      
.
13}T2024·Çý·çìUü𝑡∈RHôõ»®(1,2)ÇHÚ x &'(2𝑥>
1
𝑥+𝑡 ‰Ją‘ƨÿH t ö
¹÷øn      
}
14}T2023·û먽·ìðñòUü𝑓(𝑥)[šJõ𝑅Ç Ď)
Hą–𝑥≥0˜H𝑓(𝑥) = e𝑥H&'(𝑓(𝑥) ≥
𝑓2(𝑥−1) 5æn      
.
û95ń
15}T2023·ġČ·ìðñòU?"
(1)(−π)0 + (1.5)−2 ×
3
27
8
2−
1
0.01 +
92©
(2)
5 𝑎−3𝑏2 ÷
5 𝑎2 ÷
5 𝑏3
16}T2024·ġČŅÕ·ìðñòUT1U?"
 9
4
1
2−(−9.6)0−27
8
−2
3 +
3
2
−2
©
T2UÙy𝑎
1
2 + 𝑎−1
2 = 3H<𝑎2+𝑎−2+1
𝑎+𝑎−1+2 ¹.
17}T2024·ÇýņŇ·úìUü𝑎∈RH)
𝑓(𝑥) = 2𝑥+𝑎
2𝑥−1.
(1)<𝑎 ¹HćĈ𝑦= 𝑓(𝑥)nę)
©
(2)ô𝑓(2) = 𝑎H<āĂ𝑓(𝑥) > 𝑎 Ā
𝑥 ö¹÷ø.
18}T23-24 ~Ç·YZ¡ň·ÌÍUÙy)
𝑓(𝑥) = 𝑎⋅8𝑥+2𝑥
𝑎⋅4𝑥T𝑎n\
Hą𝑎≠0H𝑎∈RU}
      
      
(1)–𝑎= −1˜Hôrĸ‰ 𝑥∈[1,2]H½ 𝑓(2𝑥) ≥𝑚𝑓(𝑥)¨ÿH<Ā
𝑚 ö¹÷ø©
(2)–𝑓(𝑥)nĎ)
˜HôÚ𝑥 #$𝑓(2𝑥) = 𝑚𝑓(𝑥) Ā
5H<Ā
𝑚 ö¹÷ø}
19}T2024·ļÜňʼnġ·ìðñòUÙy)
𝑓(𝑥) = 1−
𝑎−1
𝑎𝑥+1(𝑎> 0ą𝑎≠1UnšJõ R Ç ę)
(1)./0, šJė¶
)
𝑓(𝑥)õ R Ç/0îÀ©
(2)ôÚ x &'(𝑓(𝑚𝑥2−1) + 𝑓(2−𝑚𝑥) > 0ƨÿH<Ā
 m ö¹÷ø©
(3)ô)
𝑔(𝑥) = 𝑘𝑓(𝑥)−3𝑥 ąŊ èóŋ:H<Ā
 k ö¹÷ø}