第二章函數(shù)第7節(jié)對(duì)數(shù)與對(duì)數(shù)函數(shù)學(xué)習(xí)導(dǎo)航站核心知識(shí)庫(kù)：重難考點(diǎn)總結(jié)，梳理必背知識(shí)、歸納重點(diǎn)考點(diǎn)1對(duì)數(shù)的概念★★☆☆☆考點(diǎn)2對(duì)數(shù)的性質(zhì)、運(yùn)算性質(zhì)與換底公式★★★☆☆考點(diǎn)3對(duì)數(shù)函數(shù)及其性質(zhì)★★★☆☆考點(diǎn)4反函數(shù)★★★☆☆（星級(jí)越高，重要程度越高）限時(shí)【變式訓(xùn)練】挑戰(zhàn)場(chǎng)：感知真題，檢驗(yàn)成果，考點(diǎn)追溯【知識(shí)梳理】考點(diǎn)1對(duì)數(shù)的概念如果ax=N(a>0,且a≠1),那么數(shù)x叫做以a為底N的對(duì)數(shù),記作x=logaN,其中a叫做對(duì)數(shù)的底數(shù),N叫做真數(shù).考點(diǎn)2對(duì)數(shù)的性質(zhì)、運(yùn)算性質(zhì)與換底公式(1)對(duì)數(shù)的性質(zhì)①alogaN=N;②logaab=b(a>0,(2)對(duì)數(shù)的運(yùn)算性質(zhì)如果a>0且a≠1,M>0,N>0,那么①loga(MN)=logaM+logaN;②logaMN=logaM-logaN③logaMn=nlogaM(n∈R).(3)換底公式:logab=logcblogca(a>0,且a≠1,b>0,考點(diǎn)3對(duì)數(shù)函數(shù)及其性質(zhì)(1)概念:函數(shù)y=logax(a>0,且a≠1)叫做對(duì)數(shù)函數(shù),其中x是自變量,定義域是(0,+∞).(2)對(duì)數(shù)函數(shù)的圖象與性質(zhì)a>10<a<1圖象性質(zhì)定義域:(0,+∞)值域:R當(dāng)x=1時(shí),y=0,即過(guò)定點(diǎn)(1,0)當(dāng)x>1時(shí),y>0;當(dāng)0<x<1時(shí),y<0當(dāng)x>1時(shí),y<0;當(dāng)0<x<1時(shí),y>0在(0,+∞)上是增函數(shù)在(0,+∞)上是減函數(shù)考點(diǎn)4反函數(shù)指數(shù)函數(shù)y=ax(a>0,且a≠1)與對(duì)數(shù)函數(shù)y=logax(a>0,且a≠1)互為反函數(shù),它們的圖象關(guān)于直線(xiàn)y=x對(duì)稱(chēng).它們的定義域和值域正好互換.【名師點(diǎn)撥】1.換底公式的兩個(gè)重要結(jié)論(1)logab=1logba(a>0,且a≠1;b>0,(2)logambn=nmlogab(a>0,且a≠1;b>0;m,n∈R,且2.對(duì)數(shù)函數(shù)y=logax(a>0,且a≠1)的圖象恒過(guò)點(diǎn)(1,0),(a,1),1a3.對(duì)數(shù)函數(shù)的圖象與底數(shù)大小的比較如圖,作直線(xiàn)y=1,則該直線(xiàn)與四個(gè)函數(shù)圖象交點(diǎn)的橫坐標(biāo)為相應(yīng)的底數(shù),故0<c<d<1<a<b.即在第一象限內(nèi)從左到右底數(shù)逐漸增大.【教材回歸】概念思考辨析+教材經(jīng)典改編1.思考辨析(在括號(hào)內(nèi)打“√”或“×”)(1)函數(shù)y=log2(x+1)是對(duì)數(shù)函數(shù).()(2)函數(shù)y=ln1+x1?x與y=ln(1+x)-ln(1(3)當(dāng)x>1時(shí),若logax>logbx,則a<b.()(4)函數(shù)y=log2x與y=log12.(北師大必修一P105【典例】4(1)改編)計(jì)算:log4259+log23-log0.515=3.(人教B必修二P27【典例】2改編)已知log0.7(2m)<log0.7(m-1),則m的取值范圍是.

4.(人教A必修一P141T13(1)改編)設(shè)a=log0.26,b=log0.36,c=log0.46,則a,b,c的大小關(guān)系為.

【考向核心題型】考點(diǎn)一對(duì)數(shù)的運(yùn)算【典例】1.(2025·丹東模擬)若2a=3,3b=5,5c=4,則log4(abc)=()A.-2 B.12C.22 【典例】2.計(jì)算:(1?log63)2【變式訓(xùn)練】1.計(jì)算:log535+2log122-log5150-log5【變式訓(xùn)練】2.已知lg2=a,lg3=b,用a,b表示log1815=.

考點(diǎn)二對(duì)數(shù)函數(shù)的圖象及應(yīng)用【典例】3.(多選)(2025·青島調(diào)研)已知ax=b-x,函數(shù)y=loga(-x)與y=bx的圖象可能是()【典例】4.已知函數(shù)f(x)=|log3x|,若a<b,且f(a)=f(b),則a+4b的取值范圍是()A.[22,+∞) B.(22,+∞)C.[5,+∞) D.(5,+∞)【變式訓(xùn)練】3.已知函數(shù)f(x)=loga(2x+b-1)(a>0且a≠1)的圖象如圖所示,則a,b滿(mǎn)足的關(guān)系是()A.0<a-1<b<1 B.0<b<a-1<1C.0<b-1<a<1 D.0<a-1<b-1<1【變式訓(xùn)練】4.若方程4x=logax在0,12上有解,則實(shí)數(shù)a的取值范圍為考點(diǎn)三對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用角度1比較大小【典例】5.(多選)(2025·南京、鹽城模擬)已知x,y∈R,且12x=3,12y=4,則()A.y>x B.x+y>1C.xy<14 D.x+y<角度2解對(duì)數(shù)方程或不等式【典例】6.已知函數(shù)f(x)=2x2,x≥0,?2x2,x<0,則不等式f((log2x角度3對(duì)數(shù)函數(shù)性質(zhì)的綜合應(yīng)用【典例】7.(多選)(2025·泉州模擬)已知函數(shù)f(x)=lgx2?A.f(x)的最小值為1B.?x∈R,f(1)+f(x)=2C.f(log92)>f2D.f90.1?1【變式訓(xùn)練】5.(2025·衡陽(yáng)質(zhì)檢)已知a=log21.8,b=log43.6,c=12,A.a>b>c B.a>c>bC.b>a>c D.b>c>a【變式訓(xùn)練】6.(2025·銀川調(diào)研)已知函數(shù)f(x)=ln(e+x)-ln(e-x),則f(x)是()A.奇函數(shù),且在(0,e)上單調(diào)遞增B.奇函數(shù),且在(0,e)上單調(diào)遞減C.偶函數(shù),且在(0,e)上單調(diào)遞增D.偶函數(shù),且在(0,e)上單調(diào)遞減【限時(shí)訓(xùn)練】（限時(shí)：60分鐘）一、單選題1.(2025·杭州質(zhì)檢)已知x,y∈R+,若3x=4y且2x=ay,則a=()A.2log32 B.log32 C.2log23 D.4log322.(2025·重慶診斷)設(shè)函數(shù)f(x)=lg(x2+1),則使得f(2x-1)>f(x+1)成立的x的取值范圍為()A.(0,2) B.(2,+∞)C.(-∞,2) D.(-∞,0)∪(2,+∞)3.若logx10.8<logx20.8<0,則A.0<x2<x1<1 B.0<x1<x2<1C.1<x1<x2 D.1<x2<x14.在同一平面直角坐標(biāo)系中,函數(shù)y=loga(-x),y=a?1x(a>0且a5.(2025·通化模擬)設(shè)a=log0.14,b=log504,則()A.2ab<2(a+b)<ab B.2ab<a+b<4abC.ab<a+b<2ab D.2ab<a+b<ab6.(2025·常州質(zhì)測(cè))設(shè)函數(shù)f(x)=log12x+ax在區(qū)間[1,+A.(-∞,1] B.[0,1]C.(-1,1] D.[1,+∞)7.已知a>0,且a≠1,則函數(shù)y=logax+A.第一、二象限 B.第一、三象限C.第二、四象限 D.第三、四象限8.若不等式(x-1)2<logax(a>0且a≠1)在x∈(1,2]內(nèi)恒成立,則實(shí)數(shù)a的取值范圍為()A.(1,2] B.(1,2)C.(1,2] D.(2,2)二、多選題9.函數(shù)f(x)=loga|x-1|(a>0,且a≠1)在(0,1)上是減函數(shù),那么()A.f(x)在(1,+∞)上單調(diào)遞增且無(wú)最大值B.f(x)在(1,+∞)上單調(diào)遞減且無(wú)最小值C.f(x)在定義域內(nèi)是偶函數(shù)D.f(x)的圖象關(guān)于直線(xiàn)x=1對(duì)稱(chēng).10.已知函數(shù)f(x)=?x2?4x,x≤0,|log2x,x>0,若x1<x2<x3<x4,且f(x1)=fA.x1+x2=-4 B.x3x4=1C.1<x4<4 D.0<x1x2x3x4≤211.(2025·河南名校聯(lián)考)已知正數(shù)x,y,z滿(mǎn)足5x=9y=15z,則()A.xz+2yz-2xy=0 B.5x<9y<15zC.xy<2z2 D.9x+2y<16z三、填空題12.(2025·1月八省聯(lián)考)已知函數(shù)f(x)=ax(a>0,a≠1),若f(ln2)f(ln4)=8,則a=.

13.函數(shù)f(x)=log2x·log2(2x)的最小值為.14.(2025·曲靖質(zhì)測(cè))如圖,在第一象限內(nèi),矩形ABCD的三個(gè)頂點(diǎn)A,B,C分別在函數(shù)y=log33x,y=x13,y=33x的圖象上,且矩形的邊分別與兩坐標(biāo)軸平行,若A點(diǎn)的縱坐標(biāo)是2,四、解答題15.已知函數(shù)f(x)=log2(1+x),g(x)=log2(1-x).(1)求函數(shù)f(x)-g(x)的定義域;(