专题2.5对数与对数函数六大题型（原卷版）_11.4-2_2026-11-04

      
      
专题2.5 对数与对数函数【六大题型】

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U 1[2024·ÛÜ·ÄÝ]Þ𝑎≥0,𝑏∈𝑅Eœ™2log23 + ( 𝑎)2 +
𝑏2{ßl[    ]
A3 + 𝑎+ 𝑏
B3 + 𝑎+ |𝑏|
C2 + 𝑎+ 𝑏
D2 + 𝑎+ |𝑏|
–5 1-1[2024·àá·Ýâãä]Þ𝑎= log35E5𝑏= 6Eœ𝑎𝑏−log32 = [   ]
A1
Bå1
C2
Då2
–5 1-2[2024·æççè·Ýâãä]é aEbEc Élê
E\4𝑎= 6𝑏= 9𝑐= 𝑡Eëì[   ]
A
1
𝑎+
1
𝑏=
1
𝑐
B
1
𝑏+
1
𝑐=
1
𝑎
C
1
𝑎+
1
𝑏=
2
𝑐
D
1
𝑎+
1
𝑐=
2
𝑏
–5 1-3[2024·íîïð·JÝ]Þ2𝑎= 3E3𝑏= 5E5𝑐= 4Eœlog4𝑎𝑏𝑐= [    ]
A−2
B
1
2
C 2
2
D1
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U 2[2024·ñòóè·ÄÝ]Íôõ9ö÷øùúú¼Eö÷—Íôõ};l21 × 2101Y
Ztû'uEÐ 441 ü:.ýþ 0 Ÿ 1 ͔ÿĀõāœEJæ2441ă3žõEĄéö÷ 1 ąaĆ
3 × 1015üÍôõEëìȦÍôõ0ćĈ~[    ][ĉ
þ
lg2 ≈0.301,lg3 ≈0.477]
A10117ąa
B10120ąa
C10123ąa
D10125ąa
–5 2-1[2024·jkĊċ·ÍÝ]‘ČčĎ0ďĐEčđĒē9čĔØŸđĕĖ¦EėĘ
ęĚEěăĜč 90℃đĝÿE¥4ĞčđĕĖğĠ 60℃­ġEĈ~ĢøģĤĒē̐ 20℃ĥĕĦE
      
      
čđĕĖr 90℃ħĨEėS tmin šĕĖ_𝑦℃EĈĩ&
𝑦= 60 × 0.9𝑡+20(𝑡≥0)wsĪīĬXčđĕ
Ėĭ­Į–āįEœ±İıIJĦEijăĜččđĴИĤġĒē­Eĵ;Ķķ­Į˜ĸsl
[    ]
[ĉ
þ
lg2 ≈0.30,lg3 ≈0.48]
A2.5min
B4.5min
C6min
D8min
–5 2-2[2024·èĹ·Ýâãä]ĺrÒTĻøùļĽ0ľ>?
þ‘Ŀŀ~ 1 ħŁ
Ľŀ
łŃ/E~ 1 ħŁ
Ľŀł
ć_ń
ÄMEŅĽÆį
0ľ b ”ÿĭņ
þ‘E~ n ħŁ

ĽŀCŃ_𝑃𝑏(𝑛) = log𝑏
𝑛+1
𝑛EŇňʼnëŊ
#ŋŌ
#ō
4É./¸8ijÆįšwQ¦
Ŏŏ
ŐőÆįwŒĘěºėœ
þ#Ŕ
þ40
þ=T-Þ
𝑘
𝑛=4𝑃10(𝑛) =
ln6−ln2
ln2+ln5[𝑘∈N∗E
𝑘> 4]Eœ k ½_[    ]
A11
B15
C19
D21
–5 2-3[2024·ŕČ·Ýâãä]ą¦ŖŗÆįlŘČř0ĺ#
ĺ#b匜ŅĽw
EÐŜ©ŝü6:RSŞşŠ¨š±ŗ¼^ŢŖEţ
ęĴ5_𝐹= 𝐺
𝑚1⋅𝑚2
𝑟2 Eţ‘𝐹ęŤŝü…Į
Ŗŗ01E𝐺_ŖŗQ
E𝑚1,𝑚2A¬ęŤŝü…6ľE𝑟ęŤŝü…ĮťŦ.Þīŧ9Ũŧs
ī:ĮťŦ_𝑟1E9Ũŧũī:ĮťŦ_𝑟2Eīŧ9Ũŧsī:ĮŖŗ01_𝐹1E9Ũŧũī:Į
Ŗŗ01_𝐹2Eœ[    ]
Aln 𝐹1 + ln𝑟1 = ln 𝐹2 + ln𝑟2
Bln 𝐹1 + ln𝑟2 = ln 𝐹2 + ln𝑟1
Cln 𝐹1 ⋅ln𝑟1 = ln 𝐹2 ⋅ln𝑟2
Dln 𝐹1 ⋅ln𝑟2 = ln 𝐹2 ⋅ln𝑟1
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&
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U 3[2024·Ūj·Ýâãä]&
𝑓(𝑥) = e𝑥−e
1
𝑥−ln𝑥2'(0ū_[   ]
A
B
      
      
C
D
–5 3-1[2024·ŕČ·Ýâãä]°‰&
𝑓(𝑥) = |𝑥|−1
|𝑥| ln𝑥2Eœ𝑓(𝑥)'(0ū_[    ]
A
B
C
D
–5 3-2[2024·æçŬŭ·ÍÝ]&
𝑓(𝑥) =
1+ln|𝑥|
𝑒𝑥+𝑒−𝑥'Ů0ū_[    ]
A
B
C
      
      
D
–5 3-3[2024·ůŰűÜ·JÝ]&
𝑓(𝑥) =
𝑥
2ln|𝑥|'Ů0ū_[    ]
A
  
B
  
C
  
D
  
 4  *
&
+,-./01
U 4[2024·bcŲá·ÄÝ]°‰𝑎= 2log20.4E𝑏= log0.42E𝑐=
1
log0.30.4Eœ[    ]
A𝑎> 𝑏> 𝑐
B𝑏> 𝑎> 𝑐
C𝑐> 𝑎> 𝑏
D𝑎> 𝑐> 𝑏
–5 4-1[2024·bcjų·ÄÝ]°‰𝑎= 0.53.1E𝑏= log0.90.3E𝑐= log1
3
1
2Eœ aEbEc 01Î_
[    ]
A𝑐< 𝑏< 𝑎
B𝑐< 𝑎< 𝑏
C𝑏< 𝑎< 𝑐
D𝑎< 𝑐< 𝑏
–5 4-2[2024·ŴŵŴŶ·ÄÝ]°‰𝑎= 40.3,𝑏= (log4𝑎)4,𝑐= log4(log4𝑎)Eœ[    ]
A𝑎> 𝑏> 𝑐
B𝑎> 𝑐> 𝑏
C𝑏> 𝑐> 𝑎
D𝑐> 𝑎> 𝑏
–5 4-3[2024·ŕČ·Ýâãä]°‰&
𝑓(𝑥)ŷŸ𝑓(𝑥) = 𝑓(2−𝑥)E\ŹĮ[1, + ∞)±+,źŻ.é
𝑎= 𝑓(−ln1.1)E𝑏= 𝑓(20.4)E𝑐= 𝑓(log25)Eœ[    ]
A𝑎> 𝑏> 𝑐
B𝑏> 𝑐> 𝑎
C𝑐> 𝑏> 𝑎
D𝑏> 𝑎> 𝑐
      
      
 5  2
345
U 5[2024·Ūj·Ýâãä]°‰&
𝑓(𝑥) = |ln𝑥|EÞ𝑓(𝑎) ≤𝑓(𝑎+ 1)EœT
 a ż½Žžl[   ]
A
5−1
2
,1
B 0,
5−1
2
C
5−1
2
, + ∞
D[1, + ∞)
–5 5-1[2024·ŕČ·Ýâãä]°‰&
𝑓(𝑥)lÆǐ R ±ſ&
Eƀ𝑥≥0­E𝑓(𝑥) = 3𝑥+2𝑥−1E
œ345𝑓(log2𝑥)−32 < 02Ɓ_[    ]
A(−∞,8)
B(0,8)
C 1
8 ,8
D 0, 1
8
–5 5-2[2024·ÛÜ·Ýâãä]“𝑎> 𝑏”l“ln(𝑎2 + e) > ln(𝑏2 + e)”[   ]
AƂ;ıIJ
BƂA3À;ıIJ
CÀ;3ƂAıIJ
Dƃ3ƂAƄ3À;ıIJ
–5 5-3[2024·ŪÜƅH·Ýâãä]°‰&
𝑓(𝑥) = 𝑎𝑥+ (𝑘−1)𝑎−𝑥[𝑎> 0\𝑎≠1]lſ&
Eœ
¯ x 345𝑓(log𝑘𝑥) > 𝑎2+1
𝑎2Ɓl[    ]
A(2, + ∞)
B 0, 1
2 ∪(2, + ∞)
C 1
2 ,2
D~±ƆƇÉ3
 6  
&
-678%
U 6[2023·ÛÜƈŶ·Ýâãä]°‰&
𝑓(𝑥)lÆǐR±Ɖ&
Eƀ𝑥< 0­E𝑓(𝑥) = log2
1+2𝑥
3−2𝑥
(1)<𝑓(𝑥)2B5Ì
(2)Þ¯𝑥¨»𝑓(𝑥) = 𝑘R±¦2E<T
𝑘ż½Žž
–5 6-1[2024·æçèƊ·JÝ]°‰&
𝑓(𝑥) = log2(𝑎𝑥2 + 4𝑥+ 5).
(1)Þ𝑓(1) = 3E<&
𝑓(𝑥)+,ŹĮÌ
(2)lƋƌT
 aE—&
𝑓(𝑥)˜1½_ 0ƍÞƌE<Ľ a ½ÌÞ3ƌEƎƏĚÒ.
      
      
–5 6-2[23-24 JĦ·ƐðƑŁ·ƒ‘]°‰&
𝑓(𝑥) =
2𝑥+1
2𝑥+𝑎_Ɖ&

(1)<T
 a ½Ì
(2)ƓƔ&
𝑓(𝑥)+,-[3ƕĚ]Ì
(3)é&
𝑔(𝑥) = log2
𝑥
2 ⋅log2
𝑥
4 +𝑚EÞŜ©𝑥1 ∈[2,8]Eńƌ𝑥2 ∈(0,1]E—ļ𝑔(𝑥1) = 𝑓(𝑥2)MƖE<T

 m ż½Žž
–5 6-3[23-24 ı·ÛjƗƘ·ŋƙƚƛ]°‰&
𝑓(𝑥) = log1
9
𝑎−𝑥
2+𝑏𝑥E𝑔(𝑥) = 𝑚⋅4𝑥−2𝑥+2 +3
(1)Þy = lg[𝑔(𝑥)]½¾_RE<ŷŸıIJƜ
𝑚½Ì
(2)ÞƝQ
&
𝑓(𝑥)lÆǾ_(−2,2)Ɖ&
E\∀𝑥1 ∈[1,2)E∃𝑥2 ∈[−1,1]E𝑓(𝑥1)−𝑔(𝑥2) > −
1
2E<𝑚
ż½Žž.
J#+
1[2024·Ɛð·ÍÝ]°‰êT
𝑚,𝑛ŷŸ
1
2ln𝑚= ln(𝑚−2𝑛)−
1
2ln𝑛Eœ
𝑛
𝑚= [    ]
A1
B
1
4
C4
D1 P
1
4
2[2024·ñò·Ýâãä]ÞT
𝑚E𝑛E𝑡ŷŸ5𝑚= 7𝑛= 𝑡\
1
𝑚+
1
𝑛= 2Eœ𝑡= [    ]
A2 3
B12
C 5
D 35
3[2024·±á·ÄÝ]°‰&
𝑓(𝑥) = 1 + log𝑎(2𝑥−3) (𝑎> 0,𝑎≠1)ƞSÆ:(𝑚,𝑛)Eœ𝑚+ 𝑛= [    ]
A1
B2
C3
D4
      
      
4[2024·gçƟƠ·Ýâãä]19 ơƢƣČbĺçƤ·ƥƊƦƧ
ę­EſOĿŀę‘~ 1 ħŁ

ĽŀłŃƨćƩüơƢšE倷ƪYƫnĿŀƬüŀ(ErTĻøùļĽ0ľ
þ‘E
~ 1 ħŁ
Ľŀł
ć_ń
ÄMEŅĽ€·ƪYÆįEА0ľ b ”ÿĭņ
þ‘E~ n ħŁ

ĽŀCŃ_𝑃𝑏(𝑛) = log𝑏
𝑛+1
𝑛EŇƭʼnëŊ
#ŋŌ
#ō
4É./¸8ijÆįEšwQ¦
ŎŏŐ
őÆįwŒĘěºėœ
þ#Ŕ
þ40
þ=T-Þ
2024
𝑛=𝑘𝑃10(𝑛) =
log29−log23
1+log25 [𝑘∈𝑁*E
𝑘≤2024]Eœ k ½_[    ]
A674
B675
C676
D677
5[2024·îƮƯưƱ·Ýâãä]°‰𝑎= log56E𝑏= log2 8E𝑐=
eEœ𝑎E𝑏E𝑐01Î_[    ]
A𝑎< 𝑏< 𝑐
B𝑎< 𝑐< 𝑏
C𝑐< 𝑎< 𝑏
D𝑏< 𝑐< 𝑎
6[2024·æçèƊ·Ýâãä]°‰&
𝑓(𝑥) =
−
1
4𝑥−4 ,𝑥≤3
4
log𝑎(4𝑥)−1,𝑥> 3
4
 
lR±+,&
EœT
𝑎ż½Žž
l[    ]
A(0,1)
B(1, 3]
C(1, 3)
D(1,3)
7[2024·Ɛç·Ýâãä]°‰&
𝑓(𝑥) = 1−22𝑥
1+22𝑥E𝑔(𝑥) = log2|𝑥|EŇ'_&
ℎ(𝑥)'(Eœℎ(𝑥)ĈF_
[    ]
Aℎ(𝑥) = 𝑓(𝑥) +𝑔(𝑥)
Bℎ(𝑥) = 𝑓(𝑥)−𝑔(𝑥)
Cℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥)
Dℎ(𝑥) =
𝑓(𝑥)
𝑔(𝑥)
8[2024·gç·ÍÝ]°‰ÆǐR±&
𝑓(𝑥)ŷŸ𝑓(𝑥+ 2) = 𝑓(−𝑥) = −𝑓(𝑥)Eƀ0 < 𝑥≤1­E𝑓(𝑥) = log2
(𝑥+ 1).Þ𝑓(𝑎+ 1) > 𝑓(𝑎)EœT
𝑎ż½Žžl[    ]
A −5
2 + 4𝑘,−3
2 + 4𝑘E𝑘∈Z
B(−1 + 4𝑘,4𝑘)E𝑘∈Z
C −1
2 + 4𝑘, 1
2 + 4𝑘E𝑘∈Z
D −3
2 + 4𝑘, 1
2 + 4𝑘E𝑘∈Z
Í#Ʋ
9[2024·ŕČ·Ýâãä]°‰𝑎= log49E𝑏=
1
2log35Eœ[    ]
      
      
A1 < 𝑎<
3
2
B𝑎𝑏> 1
C𝑎+ 𝑏> 2
D2 < 𝑎+
1
𝑎<
13
6
10[2024·ÛÜ·ÄÝ]°‰&
𝑓(𝑥) = lg(1−𝑥)Eœ[    ]
A𝑓(𝑥)ÆǾ_(−∞,1)
B𝑓(𝑥)½¾_𝑅
C𝑓(−1) +𝑓(−4) = 1
D𝑦= 𝑓(𝑥2)+,źƳŹĮ_(0,1)
11[2024·ÛÜ·JÝ]Æǐ R ±&
𝑓(𝑥) = log𝑎( 1 + 𝑏2𝑥2 +𝑏𝑥)[𝑎> 0\𝑎≠1E𝑏≠0]EÞƌT

 m —ļ345𝑓(−𝑚+
𝑚2 + 12) + 𝑓(−𝑚) ≥0ƞMƖEœĦƴƵİêƶl[    ]
AÞ𝑎> 1E𝑏> 0EœT
 m ż½Žž_[−2,2]
BÞ0 < 𝑎< 1E𝑏< 0EœT
 m ż½Žž_(−∞,2]
CÞ𝑎> 1E𝑏< 0EœT
 m ż½Žž_(−∞,−2] ∪[2, + ∞)
DÞ0 < 𝑎< 1E𝑏> 0EœT
 m ż½Žž_[2, + ∞)
Ä#ƷƸ
12[2024·±á·Ýâãä]°‰êT
𝑎,𝑏ŷŸlog𝑎𝑏+ log𝑏𝑎=
5
2E𝑎𝑎= 𝑏𝑏Eœ𝑎+ 𝑏=      
.
13[2024·ƹƺ·Ýâãä]Þ&
𝑓(𝑥) = ln(𝑎𝑥+ 1)(1,2)±+,źŻEœT
𝑎ż½Žž_      

14[2024·æççè·Ýâãä]&
𝑦= log𝑎𝑥+ 𝑎𝑥−1 +2[𝑎> 0\𝑎≠1]'(ƞSÆ:(𝑘,𝑏)EÞ
𝑚+ 𝑛= 𝑏−𝑘\𝑚> 0E𝑛> 0Eœ
9
𝑚+
1
𝑛˜1½_      

ñ#2Ɔ
15[2023·gƻŞƼƽ·Ýâãä]?!
(1) 2 7
9
0.5
+ 0.1−2 + 2 10
27
−2
3−3π0 +
37
48Ì
(2)log23 ⋅log34 + (lg5)2 + lg5 ⋅lg20 +
1
2lg16−2log23.
16[2023·ƹƺƾƿ·Ýâãä][1]<½
(
3 2 ×
3)
6 + ( 2 2)
4
3−4 ×
16
49
−1
2−
4 2 × 80.25−(−2023)0Ì
[2]°‰lg𝑥+ lg𝑦= 2lg(𝑥−2𝑦)E<log 2
𝑥
𝑦½.
      
      
17[2023·ŪÜǀŶ·Ýâãä]°‰&
𝑓(𝑥) = −(log2𝑥)2 +𝑚log4𝑥+ 2 (2 ≤𝑥≤16)E\&
𝑓(𝑥)'(
ėS:(4,6).
(1)<T
𝑚½Ì
(2)<&
𝑓(𝑥)˜1½Ÿ˜0½.
18[2024·æçǁƺ·JÝ]°‰𝑓(𝑥)lÆǐ𝑅±ſ&
E\𝑥≤0­E𝑓(𝑥) = log1
2(−𝑥+ 1).
(1)<𝑓(𝑥)2B5Ì
(2)Þ𝑓(𝑎−1)−𝑓(1) < 0E<T
𝑎ż½Žž.
19[23-24 ı·ŕČ·ŋƙƚƛ]°‰&
𝑓(𝑥) = log2(2𝑥+ 1) +𝑎𝑥lſ&
.
(1)<𝑎½;
(2)é𝑔(𝑥) = 𝑓(𝑥) + 𝑥 Eℎ(𝑥) = 𝑥2−2𝑥+ 𝑚EÞŜ©𝑥1 ∈[0,4] Eƌ𝑥2 ∈[0,5]E—ļ𝑔(𝑥1) ≥ℎ(𝑥2)E
<𝑚ż½Žž.