新高考數(shù)學(xué)一輪復(fù)習(xí)講練測第2章第05講 對數(shù)與對數(shù)函數(shù)（講義）（原卷版）

第05講對數(shù)與對數(shù)函數(shù)1、對數(shù)式的運(yùn)算(1)對數(shù)的定義：一般地，如果SKIPIF1<0且SKIPIF1<0，那么數(shù)SKIPIF1<0叫做以SKIPIF1<0為底SKIPIF1<0的對數(shù)，記作SKIPIF1<0，讀作以SKIPIF1<0為底SKIPIF1<0的對數(shù)，其中SKIPIF1<0叫做對數(shù)的底數(shù)，SKIPIF1<0叫做真數(shù)．(2)常見對數(shù)：①一般對數(shù)：以SKIPIF1<0且SKIPIF1<0為底，記為SKIPIF1<0，讀作以SKIPIF1<0為底SKIPIF1<0的對數(shù)；②常用對數(shù)：以SKIPIF1<0為底，記為SKIPIF1<0；③自然對數(shù)：以SKIPIF1<0為底，記為SKIPIF1<0；(3)對數(shù)的性質(zhì)和運(yùn)算法則：①SKIPIF1<0；SKIPIF1<0；其中SKIPIF1<0且SKIPIF1<0；②SKIPIF1<0(其中SKIPIF1<0且SKIPIF1<0，SKIPIF1<0)；③對數(shù)換底公式：SKIPIF1<0；④SKIPIF1<0；⑤SKIPIF1<0；⑥SKIPIF1<0，SKIPIF1<0；⑦SKIPIF1<0和SKIPIF1<0；⑧SKIPIF1<0；2、對數(shù)函數(shù)的定義及圖像(1)對數(shù)函數(shù)的定義：函數(shù)SKIPIF1<0SKIPIF1<0且SKIPIF1<0叫做對數(shù)函數(shù)．對數(shù)函數(shù)的圖象SKIPIF1<0SKIPIF1<0圖象性質(zhì)定義域：SKIPIF1<0值域：SKIPIF1<0過定點(diǎn)SKIPIF1<0，即SKIPIF1<0時(shí)，SKIPIF1<0在SKIPIF1<0上增函數(shù)在SKIPIF1<0上是減函數(shù)當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0【解題方法總結(jié)】1、對數(shù)函數(shù)常用技巧在同一坐標(biāo)系內(nèi)，當(dāng)SKIPIF1<0時(shí)，隨SKIPIF1<0的增大，對數(shù)函數(shù)的圖象愈靠近SKIPIF1<0軸；當(dāng)SKIPIF1<0時(shí)，對數(shù)函數(shù)的圖象隨SKIPIF1<0的增大而遠(yuǎn)離SKIPIF1<0軸．(見下圖)題型一：對數(shù)運(yùn)算及對數(shù)方程、對數(shù)不等式【例1】（2023·四川成都·成都七中?？寄M預(yù)測）SKIPIF1<0______．【對點(diǎn)訓(xùn)練1】（2023·遼寧沈陽·沈陽二中?？寄M預(yù)測）已知SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0______．【對點(diǎn)訓(xùn)練2】（2023·上海徐匯·位育中學(xué)校考模擬預(yù)測）方程SKIPIF1<0的解集為________．【對點(diǎn)訓(xùn)練3】（2023·山東淄博·統(tǒng)考二模）設(shè)SKIPIF1<0，滿足SKIPIF1<0，則SKIPIF1<0__________.【對點(diǎn)訓(xùn)練4】（2023·天津南開·統(tǒng)考二模）計(jì)算SKIPIF1<0的值為______.【對點(diǎn)訓(xùn)練5】（2023·全國·高三專題練習(xí)）若SKIPIF1<0，SKIPIF1<0，用a，b表示SKIPIF1<0____________【對點(diǎn)訓(xùn)練6】（2023·上?！じ呷Ｂ?lián)考階段練習(xí)）若SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0__________．【對點(diǎn)訓(xùn)練7】（2023·全國·高三專題練習(xí)）SKIPIF1<0=____________；【對點(diǎn)訓(xùn)練8】（2023·全國·高三專題練習(xí)）解關(guān)于x的不等式SKIPIF1<0解集為_____．【對點(diǎn)訓(xùn)練9】（2023·上海楊浦·高三上海市楊浦高級(jí)中學(xué)?？奸_學(xué)考試）已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0的解集是__________.【對點(diǎn)訓(xùn)練10】（2023·上海浦東新·高三華師大二附中校考階段練習(xí)）方程SKIPIF1<0的解為_________．【解題方法總結(jié)】對數(shù)的有關(guān)運(yùn)算問題要注意公式的順用、逆用、變形用等.對數(shù)方程或?qū)?shù)不等式問題是要將其化為同底，利用對數(shù)單調(diào)性去掉對數(shù)符號(hào)，轉(zhuǎn)化為不含對數(shù)的問題，但這里必須注意對數(shù)的真數(shù)為正.題型二：對數(shù)函數(shù)的圖像【例2】（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0（a，b為常數(shù)，其中SKIPIF1<0且SKIPIF1<0）的圖象如圖所示，則下列結(jié)論正確的是（

）A．SKIPIF1<0，SKIPIF1<0 B．SKIPIF1<0，SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<0【對點(diǎn)訓(xùn)練11】（2023·全國·高三專題練習(xí)）函數(shù)SKIPIF1<0的圖象恒過定點(diǎn)（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【對點(diǎn)訓(xùn)練12】（2023·北京·統(tǒng)考模擬預(yù)測）已知函數(shù)SKIPIF1<0，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【對點(diǎn)訓(xùn)練13】（2023·北京·高三統(tǒng)考學(xué)業(yè)考試）將函數(shù)SKIPIF1<0的圖象向上平移1個(gè)單位長度，得到函數(shù)SKIPIF1<0的圖象，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【對點(diǎn)訓(xùn)練14】（2023·北京海淀·清華附中?？寄M預(yù)測）不等式SKIPIF1<0的解集為__________．【對點(diǎn)訓(xùn)練15】（多選題）（2023·全國·高三專題練習(xí)）當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0的值可以為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【解題方法總結(jié)】研究和討論題中所涉及的函數(shù)圖像是解決有關(guān)函數(shù)問題最重要的思路和方法.圖像問題是數(shù)和形結(jié)合的護(hù)體解釋.它為研究函數(shù)問題提供了思維方向.題型三：對數(shù)函數(shù)的性質(zhì)（單調(diào)性、最值（值域））【例3】（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，若SKIPIF1<0在SKIPIF1<0上為減函數(shù)，則a的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【對點(diǎn)訓(xùn)練16】（2023·新疆阿勒泰·統(tǒng)考三模）正數(shù)SKIPIF1<0滿足SKIPIF1<0，則a與SKIPIF1<0大小關(guān)系為______．【對點(diǎn)訓(xùn)練17】（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0在SKIPIF1<0上的最大值是2，則a等于_________【對點(diǎn)訓(xùn)練18】（2023·全國·高三專題練習(xí)）若函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）在SKIPIF1<0上的最大值為2，最小值為m，函數(shù)SKIPIF1<0在SKIPIF1<0上是增函數(shù)，則SKIPIF1<0的值是____________．【對點(diǎn)訓(xùn)練19】（2023·全國·高三專題練習(xí)）若函數(shù)SKIPIF1<0有最小值，則SKIPIF1<0的取值范圍是______．【對點(diǎn)訓(xùn)練20】（2023·河南·校聯(lián)考模擬預(yù)測）寫出一個(gè)同時(shí)具有下列性質(zhì)①②③的函數(shù):SKIPIF1<0_____.①SKIPIF1<0;②當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0單調(diào)遞減;③SKIPIF1<0為偶函數(shù).【對點(diǎn)訓(xùn)練21】（2023·重慶渝中·高三重慶巴蜀中學(xué)?？茧A段練習(xí)）函數(shù)SKIPIF1<0的單調(diào)遞區(qū)間為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【對點(diǎn)訓(xùn)練22】（2023·陜西寶雞·統(tǒng)考二模）已知函數(shù)SKIPIF1<0，則（

）A．SKIPIF1<0在SKIPIF1<0單調(diào)遞減，在SKIPIF1<0單調(diào)遞增 B．SKIPIF1<0在SKIPIF1<0單調(diào)遞減C．SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對稱 D．SKIPIF1<0有最小值，但無最大值【對點(diǎn)訓(xùn)練23】（2023·全國·高三專題練習(xí)）若函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)，則a的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【解題方法總結(jié)】研究和討論題中所涉及的函數(shù)性質(zhì)是解決有關(guān)函數(shù)問題最重要的思路和方法.性質(zhì)問題是數(shù)和形結(jié)合的護(hù)體解釋.它為研究函數(shù)問題提供了思維方向.題型四：對數(shù)函數(shù)中的恒成立問題【例4】（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，SKIPIF1<0，若存在SKIPIF1<0，任意SKIPIF1<0，使得SKIPIF1<0，則實(shí)數(shù)SKIPIF1<0的取值范圍是___________.【對點(diǎn)訓(xùn)練24】（2023·全國·高三專題練習(xí)）若SKIPIF1<0，不等式SKIPIF1<0恒成立，則實(shí)數(shù)SKIPIF1<0的取值范圍為___________.【對點(diǎn)訓(xùn)練25】（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，SKIPIF1<0，對任意的SKIPIF1<0，SKIPIF1<0，SKIPIF1<0有SKIPIF1<0恒成立，則實(shí)數(shù)SKIPIF1<0的取值范圍是___________.【對點(diǎn)訓(xùn)練26】（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，若對SKIPIF1<0，使得SKIPIF1<0，則實(shí)數(shù)SKIPIF1<0的取值范圍為___________.【對點(diǎn)訓(xùn)練27】（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0，求a的值；(2)若對任意的SKIPIF1<0，SKIPIF1<0恒成立，求SKIPIF1<0的取值范圍.【對點(diǎn)訓(xùn)練28】（2023·全國·高三專題練習(xí)）已知SKIPIF1<0，SKIPIF1<0.（1）當(dāng)SKIPIF1<0時(shí)，求函數(shù)SKIPIF1<0的值域；（2）對任意SKIPIF1<0，其中常數(shù)SKIPIF1<0，不等式SKIPIF1<0恒成立，求實(shí)數(shù)SKIPIF1<0的取值范圍.【解題方法總結(jié)】（1）利用數(shù)形結(jié)合思想，結(jié)合對數(shù)函數(shù)的圖像求解；（2）分離自變量與參變量，利用等價(jià)轉(zhuǎn)化思想，轉(zhuǎn)化為函數(shù)的最值問題.（3）涉及不等式恒成立問題，將給定不等式等價(jià)轉(zhuǎn)化，借助同構(gòu)思想構(gòu)造函數(shù)，利用導(dǎo)數(shù)探求函數(shù)單調(diào)性、最值是解決問題的關(guān)鍵.題型五：對數(shù)函數(shù)的綜合問題【例5】（多選題）（2023·湖北·黃岡中學(xué)校聯(lián)考模擬預(yù)測）已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則以下結(jié)論正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【對點(diǎn)訓(xùn)練29】（2023·海南?？凇そy(tǒng)考模擬預(yù)測）已知正實(shí)數(shù)SKIPIF1<0，SKIPIF1<0滿足：SKIPIF1<0，則SKIPIF1<0的最小值為______.【對點(diǎn)訓(xùn)練30】（多選題）（2023·廣東惠州·統(tǒng)考一模）若SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【對點(diǎn)訓(xùn)練31】（2023·河南·高三信陽高中校聯(lián)考階段練習(xí)）已知SKIPIF1<0，SKIPIF1<0分別是方程SKIPIF1<0和SKIPIF1<0