重难点03指、对、幂数比较大小问题八大题型（原卷版）_11.4-2_2026-11-04

      
      
重难点03 指、对、幂数的大小比较问题【八大题型】

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Å 1Æ2024·ÇÈÉÊ·ËÌÍÎÏÐE𝑎= 30.3H𝑏= 0.33H𝑐= log0.33HÑ𝑎H𝑏H𝑐 %&”@Æ   Ï
Až𝑎> 𝑏> 𝑐
Bž𝑏> 𝑎> 𝑐
Cž𝑐> 𝑏> 𝑎
Dž𝑐> 𝑎> 𝑏
7d 1-1Æ2024·ÒÓÔÕ·ºËÏÐE𝑎= log2
1
3H𝑏= 1.20.2H𝑐= 0.52.1HÑ aHbHc %&”@Æ    Ï
Až𝑎< 𝑐< 𝑏
Bž𝑐< 𝑎< 𝑏
Cž𝑐< 𝑏< 𝑎
Dž𝑎< 𝑏< 𝑐
7d 1-2Æ2024·Ö×ÖÊ·ºËÏÐE𝑎= 40.3,𝑏= (log4𝑎)4,𝑐= log4(log4𝑎)HÑÆ    Ï
Až𝑎> 𝑏> 𝑐
Bž𝑎> 𝑐> 𝑏
Cž𝑏> 𝑐> 𝑎
Dž𝑐> 𝑎> 𝑏
7d 1-3Æ2024·ØÙÚÛ·ËÌÍÎÏÐE𝑎= log0.20.3H𝑏= ln𝑎H𝑐= 2𝑎HÑ𝑎,𝑏,𝑐 %&”ÜÆ    Ï
Až𝑐> 𝑏> 𝑎
Bž𝑎> 𝑏> 𝑐
Cž𝑏> 𝑎> 𝑐
Dž𝑐> 𝑎> 𝑏
 2  '()*#$%&
Å 2Æ23-24 ºÝ·ÞßÈà·áâãäÏÐE𝑎= e0.1H𝑏= 1−2lg2H𝑐= 2−log310HÑ aHbHc %&
”@Æ   Ï
Až𝑏> 𝑐> 𝑎
Bž𝑎> 𝑏> 𝑐
Cž𝑎> 𝑐> 𝑏
Dž𝑏> 𝑎> 𝑐
7d 2-1Æ2024·åæçÓ·ËÌÍÎÏÐE𝑎=
1
e
−1
2,𝑏= log65,𝑐= log56HÑÆ    Ï
Až𝑎< 𝑏< 𝑐
Bž𝑐< 𝑏< 𝑎
Cž𝑏< 𝑐< 𝑎
Dž𝑎< 𝑐< 𝑏
7d 2-2Æ2024·ØÙèé·°ËÏÐE𝑎= e−1H𝑏= lg𝑎H𝑐= e0HÑÆ   Ï
Až𝑏< 𝑎< 𝑐
Bž𝑏< 𝑐< 𝑎
Cž𝑎< 𝑏< 𝑐
Dž𝑐< 𝑏< 𝑎
7d 2-3Æ2024·Þßêë·ºËÏÐE𝑎= 0.53.1H𝑏= log0.90.3H𝑐= log1
3
1
2HÑ aHbHc %&”Ü
      
      
Æ    Ï
Až𝑐< 𝑏< 𝑎
Bž𝑐< 𝑎< 𝑏
Cž𝑏< 𝑎< 𝑐
Dž𝑎< 𝑐< 𝑏
 3  +,)*#$%&
Å 3Æ2024·åæ0ì·ËÌÍÎÏí𝑎= log0.50.6H𝑏= 0.49−0.3H𝑐= 0.6−0.6HÑ𝑎H𝑏H𝑐 %&”@
Æ    Ï
Až𝑐> 𝑏> 𝑎
Bž𝑏> 𝑎> 𝑐
Cž𝑏> 𝑐> 𝑎
Dž𝑐> 𝑎> 𝑏
7d 3-1Æ23-24 °€·îÈïð·ñ'ÏÐEò
𝑎,𝑏,𝑐óô2𝑎+𝑎= 2H2𝑏+𝑏=
5H𝑐= log163HÑ
Æ    Ï
Až𝑐< 𝑎< 𝑏
Bž𝑎< 𝑏< 𝑐
Cž𝑎< 𝑐< 𝑏
Dž𝑏< 𝑐< 𝑎
7d 3-2Æ2024·õö÷Ó·°ËÏø𝑎= log1
3
1
4H𝑏= ( 1
3 )
1
4H𝑐= log3
1
4H𝑑=
1
4ÑÆ    Ï
Až𝑎> 𝑏> 𝑑> 𝑐Bž𝑎> 𝑏> 𝑐> 𝑑
Cž𝑏> 𝑑> 𝑎> 𝑐
Dž𝑎> 𝑑> 𝑏> 𝑐
7d 3-3Æ2024·Þß^ù·AËÏí 1
3
𝑎
= 2,𝑏= log1
23−log1
29,𝑐=
1
2
−1
3HÑúÆ    Ï
Až𝑎< 𝑏< 𝑐
Bž𝑎< 𝑐< 𝑏
Cž𝑏< 𝑐< 𝑎
Dž𝑏< 𝑎< 𝑐
 4  -.*/-0*#$%&
Å 4Æ2023·ÒÓxt·AËÏø𝑎= 3−1
4H𝑏=
3
2
−1
3H𝑐= log1
2
2
5HÑ𝑎H𝑏H𝑐 %&”ÜÆ    Ï
Až𝑎> 𝑏> 𝑐
Bž𝑏> 𝑐> 𝑎
Cž𝑐> 𝑎> 𝑏
Dž𝑐> 𝑏> 𝑎
7d 4-1Æ2023·Ö×ûüý·ËÌÍÎÏø𝑎=
ln2
2 H𝑏=
ln3
3 H𝑐=
ln5
5 HÑÆ    Ï
Až𝑎< 𝑏< 𝑐
Bž𝑐< 𝑏< 𝑎
Cžc<a<b
Dž𝑎< 𝑐< 𝑏
7d 4-2Æ2024·ÒÓxt·°ËÏø𝑎= ln26,𝑏= 4ln2 ⋅ln3,𝑐= (1 + ln3)2HÑ𝑎,𝑏,𝑐 %&”@Æ    Ï
Až𝑐< 𝑎< 𝑏
Bž𝑎< 𝑏< 𝑐
Cž𝑐< 𝑏< 𝑎
Dž𝑏< 𝑎< 𝑐
7d 4-3Æ2024·þÿ·ËÌÍÎÏø𝑎= 20.4,𝑏= 30.25,𝑐= log0.70.5HÑ𝑎,𝑏,𝑐 %&”ÜÆ    Ï
Až𝑎< 𝑏< 𝑐
Bž𝑏< 𝑎< 𝑐
Cžb < 𝑐< a
Dž𝑐< 𝑎< 𝑏
 5  12
*#$%&
Å 5Æ2024·þÿ·ËÌÍÎÏÐE𝑎= ln
7
2H𝑏= ln7 × ln2H𝑐=
ln7
ln2HÑÆ    Ï
Až𝑏< 𝑐< 𝑎
Bž𝑏< 𝑎< 𝑐
Cž𝑎< 𝑏< 𝑐
Dž𝑎< 𝑐< 𝑏
7d 5-1Æ2024·þÿ·ËÌÍÎÏí𝑎= 5
1
4H𝑏=
5
4H𝑐= log45HÑ aHbHc %&”ÜÆ    Ï
      
      
Až𝑎> 𝑏> 𝑐
Bž𝑎> 𝑐> 𝑏
Cž𝑏> 𝑎> 𝑐
Dž𝑏> 𝑐> 𝑎
7d 5-2Æ2024·Þß^ù·AËÏÐE𝑎= log0.20.3,𝑏= log0.30.2,𝑐= log23HÑ𝑎,𝑏,𝑐 %&”ÜÆ    Ï
Až𝑏< 𝑐< 𝑎
Bž𝑐< 𝑏< 𝑎
Cž𝑎< 𝑏< 𝑐
Dž𝑎< 𝑐< 𝑏
7d 5-3Æ2023·ĀÈ·āĂËÌÍÎÏÐEò
𝑎,𝑏,𝑐óô𝑎2 + log2𝑎= 0,2023−𝑏= log2023𝑏,𝑐= log7 6H
ÑÆ     Ï
Až𝑎< 𝑏< 𝑐
Bž𝑐< 𝑎< 𝑏
Cž𝑏< 𝑐< 𝑎
Dž𝑐< 𝑏< 𝑎
 6  
345#$%&
Å 6Æ2024·ĀÈ·ËÌÍÎÏÐE𝑎= ln𝜋,𝑏= log3𝜋,𝑐=
𝜋ln2HÑ𝑎,𝑏,𝑐 %&”@Æ    Ï
Až𝑏< 𝑎< 𝑐
Bž𝑎< 𝑏< 𝑐
Cž𝑐< 𝑏< 𝑎
Džb<c<a
7d 6-1Æ2023·ăæĄ×·°ËÏølog3𝑥= log4𝑦= log5𝑧< −1HÑÆ    Ï
Až3𝑥< 4𝑦< 5𝑧Bž4𝑦< 3𝑥< 5𝑧
Cž4𝑦< 5𝑧< 3𝑥
Dž5𝑧< 4𝑦< 3𝑥
7d 6-2Æ2024·þÿ·ËÌÍÎÏÐE𝑎=
1
2
𝑎
, 1
2
𝑏
= log𝑎𝑏,𝑎𝑐= log1
2𝑐HÑò
𝑎,𝑏,𝑐 %&”ÜÆ    Ï
Až𝑎< 𝑏< 𝑐
Bž𝑎< 𝑐< 𝑏
Cž𝑐< 𝑏< 𝑎
Dž𝑐< 𝑎< 𝑏
7d 6-3Æ2024·ąÙĆć·ĈAËÏÐE𝑥,𝑦,𝑧ÃÜ%£ 0 ò
Hĉ2𝑥= 3𝑦= log5𝑧HÑ𝑥,𝑦,𝑧%&”
Ċċ @Æ    Ï
Až𝑥> 𝑦> 𝑧
Bž𝑥> 𝑧> 𝑦
Cž𝑧> 𝑥> 𝑦
Dž𝑧> 𝑦> 𝑥
 7  6789#$%&
Å 7Æ23-24 ºÝ·ÞßČč·áâãäÏí𝑎､𝑏､𝑐t@Ċ
Hĉ4𝑎= 6𝑏= 9𝑐HрĎ4ďĐđ @
Æ    Ï
Až𝑐< 𝑏< 𝑎
Bž𝑎𝑏+ 𝑏𝑐= 𝑎𝑐
Cž4𝑏⋅9𝑏= 4𝑎⋅9𝑐
Dž
1
𝑐=
2
𝑏−
1
𝑎
7d 7-1Æ2024·ăæ·ËÌÍÎÏø𝑎e𝑎= 𝑏ln𝑏(𝑎> 0)HÑÆ   Ï
Až𝑎< 𝑏
Bž𝑎= 𝑏
Cž𝑎> 𝑏
DžĒ*ċ
7d 7-2Æ2023·þÿ·ËÌÍÎÏÐE𝑎,𝑏,𝑐Ã܅ģ 1 Ċò
Hĉln𝑐= 𝑎ln𝑏,ln𝑎= 𝑏ln𝑐HÑ𝑎,𝑏,𝑐 
%&”@Æ    Ï
      
      
Až𝑐> 𝑎> 𝑏
Bž𝑏> 𝑐> 𝑎
Cž𝑎> 𝑏> 𝑐
Dž𝑎> 𝑐> 𝑏
7d 7-3Æ2024·þÿ·ËÌÍÎÏÐEĊò
𝑎H𝑏H𝑐óôe𝑐+ e−2𝑎= e𝑎+ e−𝑐H𝑏= log23 + log86H
𝑐+ log2𝑐= 2HÑ aHbHc %&”Ü(    )
Až𝑎< 𝑏< 𝑐
Bž𝑎< 𝑐< 𝑏
Cž𝑐< 𝑎< 𝑏
Dž𝑐< 𝑏< 𝑎
 8  :;*#$%&
Å 8Æ2024·åææÛ·ËÌÍÎÏø𝑎= 0.311.5,𝑏= log312,𝑐= log26,𝑑=
3 −2
3HÑúÆ    Ï
Až𝑎> 𝑏> 𝑐
Bž𝑏> 𝑎> 𝑑
Cž𝑐> 𝑎> 𝑏
Dž𝑏> 𝑐> 𝑎
7d 8-1Æ2023·ĀÈē×·ËÌÍÎÏÐE𝑎= log35H𝑏= 2 1
3
1
4H𝑐= 3log72 + log87HÑÆ    Ï
Až𝑎> 𝑏> 𝑐
Bž𝑐> 𝑏> 𝑎
Cž𝑏> 𝑎> 𝑐
Dž𝑐> 𝑎> 𝑏
7d 8-2Æ2023 Ý·ÛĔ·°āĂáâãäÏÐE𝑎=
19−17,𝑏= 6−3
4,𝑐= log53−
2
9log35HÑÆ    Ï
Až𝑎< 𝑏< 𝑐
Bžb<c<a
Cž𝑏< 𝑎< 𝑐
Džc<a<b
7d 8-3Æ2024·þÿ·ËÌÍÎÏÐE𝑎= log8.14H𝑏= log3.1eH𝑐= ln2.1,HÑÆ    Ï
Až𝑎< 𝑐< 𝑏
Bž𝑎< 𝑏< 𝑐
Cžc<a<b
Džb<c<a
A/p
1žÆ2024·þÿ·ËÌÍÎÏí𝑎= log62H𝑏= log123H𝑐= log405HÑÆ    Ï
Až𝑎< 𝑏< 𝑐
Bž𝑏< 𝑎< 𝑐
Cž𝑐< 𝑎< 𝑏
Dž𝑎< 𝑐< 𝑏
2žÆ2024·ÛĔĕ×·AËÏÐE3𝑚= 4H𝑎= 2𝑚−3H𝑏= 4𝑚−5HÑÆ    Ï
Až𝑎> 0 > 𝑏
Bž𝑏> 0 > 𝑎
Cž𝑎> 𝑏> 0
Dž𝑏> 𝑎> 0
3žÆ2024·Ö×Ėė·AËÏÐE𝑎= 3log83H𝑏= −
1
2log1
316H𝑐= log43HÑ𝑎H𝑏H𝑐 %&”ÜÆ    Ï
Až𝑎> 𝑏> 𝑐
Bž𝑐> 𝑎> 𝑏
Cž𝑏> 𝑐> 𝑎
Dž𝑏> 𝑎> 𝑐
4žÆ2023·TĘęĚě·ËÌÍÎÏí𝑎=
3
2
0.7
H𝑏=
2
3
0.7
H𝑐= log3
4(log34)HÑÆ    Ï
      
      
Až𝑐< 𝑏< 𝑎
Bž𝑎< 𝑏< 𝑐
Cž𝑐< 𝑎< 𝑏
Dž𝑎< 𝑐< 𝑏
5žÆ2024·îÈĜĝ·ËÌÍÎÏÐE𝑎= e
1
3H𝑏= ln2H𝑐= log32HÑ𝑎,𝑏,𝑐 %&”ÜÆ    Ï
Až𝑎> 𝑐> 𝑏
Bž𝑎> 𝑏> 𝑐
Cž𝑏> 𝑐> 𝑎
Dž𝑐> 𝑏> 𝑎
6žÆ2024·åæĞğ·AËÏÐEò
𝑎,𝑏,𝑐óôe2𝑎
2 = e3𝑏
3 = e5𝑐
5 = 2HÑÆ    Ï
Až𝑎> 𝑏> 𝑐
Bž𝑎< 𝑏< 𝑐
Cž𝑏> 𝑎> 𝑐
Dž𝑐> 𝑎> 𝑏
7žÆ2023·ÇÈĠ×·AËÏÐE𝑎= log3π,𝑏=
1
log3π−1,𝑐=
1
2−log3πHÑÆ    Ï
Až𝑎< 𝑏< 𝑐
Bž𝑏< 𝑐< 𝑎
Cž𝑐< 𝑎< 𝑏
Dž𝑎< 𝑐< 𝑏
8žÆ2023·åææÛ·AËÏÐE
𝑓(𝑥) = −2𝑥Hø2𝑎= log2𝑏= 𝑐HÑÆ    Ï
Až𝑓(𝑏) < 𝑓(𝑐) < 𝑓(𝑎)
Bž𝑓(𝑎) < 𝑓(𝑏) < 𝑓(𝑐)
Cž𝑓(𝑎) < 𝑓(𝑐) < 𝑓(𝑏)
Dž𝑓(𝑐) < 𝑓(𝑏) < 𝑓(𝑎)
°/Š
9žÆ2024·ĀÈìÊ·ËÌÍÎπĎĊċ @Æ    Ï
Až2−0.01 > 2−0.001
Bžlog2 3 > log2π−1
Cžlog1.85 < log1.75
Džlog33.01 > e−0.01
10žÆ2024·Cġ·ËÌÍÎÏø𝑏> 𝑐> 1H0 < 𝑎< 1HрĎ4ďĊċ @Æ    Ï
Až𝑏𝑎< 𝑐𝑎
Bžlog𝑏𝑎> log𝑐𝑎
Cž𝑐𝑏𝑎< 𝑏𝑐𝑎
Dž𝑏log𝑐𝑎> 𝑐log𝑏𝑎
11žÆ2024·Cġ·AËÏÐE3𝑎= 5𝑏= 15HрĎ4ďĊċ @Æ    Ï
Ažlg𝑎> lg𝑏
Bž𝑎+ 𝑏= 𝑎𝑏
Cž 1
2
𝑎
>
1
2
𝑏
Dž𝑎+ 𝑏> 4
º/bc
12žÆ2023·êĢģù·°ËÏ3−2,2
1
3,log25ºB
'Á% 
@      
.
13žÆ2024·êĢĤ×·ºËÏÐE𝑎= 2−1.1H𝑏= log1
4
1
3H𝑐= log23HѺ%&”Ü      
ÆĥK&Ħ%
ħĨÏ
14žÆ2023·ĩĪīĬ·ËÌÍÎÏÐE𝑎= log 3
3
2
2 H𝑏=
2
2
−3
3 H𝑐= ln
1
eHÑ aHbHc %&”Ü      
ž
Ò/gĭ
      
      
15žÆ23-24 A·þÿ·ĮįãäÏÐE𝑥= lnπH𝑦= log52H𝑧= e−1
2ž
(1)#$ xHy %&†
(2)#$ yHz %&ž
16žÆ23-24 º·þÿ·=İÏÆ1Ï#$𝑎𝑎𝑏𝑏?𝑏𝑎𝑎𝑏(𝑎> 0,𝑏> 0) %&†
Æ2ÏÐE𝑎> 2H#$log(𝑎−1)𝑎?log𝑎(𝑎+ 1)%&
17žÆ23-24 A·ÇÈ·ı{-IJÏ#$𝑎H𝑏H𝑐 %&
(1)ÐE1 < 𝑥< 2H𝑎= (log2𝑥)2H𝑏= log2𝑥2H𝑐= log2(log2𝑥)†
(2)ÐE𝑎= log36H𝑏= log510H𝑐= log714ž
18žÆ23-24 AÝ·ąÙăij·áâãäÏÐEĊò
 xHyHz óô3𝑥= 4𝑦= 6𝑧ž
Æ1ÏjĴ
1
𝑧−
1
𝑥=
1
2𝑦†
Æ2Ï#$3𝑥,4𝑦,6𝑧 %&ž
19žÆ23-24 AÝ·ąÙą×·áâãäÏÐE
𝑓(𝑥) =
𝑥2
𝑥2+1
      
      
(1)’“ĵĴĝ
𝑓(𝑥)¢®((0, + ∞)Ý pq!†
(2)ÐE𝑎= 𝑓(20.5),𝑏= 𝑓(log25),𝑐= 𝑓(0.25)HĶ#$ºB
 aHbHc %&Hĵķĝĸž