對(duì)數(shù)公式及對(duì)數(shù)函數(shù)的總結(jié)

1、對(duì)數(shù)運(yùn)算和對(duì)數(shù)函數(shù)對(duì)數(shù)的定義①若，則叫做以為底的對(duì)數(shù)，記作，其中叫做底數(shù)，叫做真(01)xaNaa???且xaNlogaxN?aN數(shù)②負(fù)數(shù)和零沒(méi)有對(duì)數(shù)。③對(duì)數(shù)式與指數(shù)式的互化：。log(010)xaxNaNaaN??????常用對(duì)數(shù)與自然對(duì)數(shù)常用對(duì)數(shù)：，即；自然對(duì)數(shù)：，即（其中…）lgN10logNlnNlogeN2.71828e?對(duì)數(shù)函數(shù)及其性質(zhì)函數(shù)名稱對(duì)數(shù)函數(shù)定義函數(shù)且叫做對(duì)數(shù)函數(shù)log(0ayxa??1)a?1a?01a??圖象定

2、義域(0)??值域R過(guò)定點(diǎn)圖象過(guò)定點(diǎn)，即當(dāng)時(shí)，(10)1x?0y?奇偶性非奇非偶單調(diào)性在上是增函數(shù)(0)??在上是減函數(shù)(0)??函數(shù)值的變化情況log0(1)log0(1)log0(01)aaaxxxxxx???????log0(1)log0(1)log0(01)aaaxxxxxx???????變化對(duì)圖象的影響a在第一象限內(nèi)，越大圖象越靠低；在第四象限內(nèi)，越大圖象越靠高。aaxyO(10)1x?logayx?xyO(10)1x?log

3、ayx?（3）對(duì)數(shù)函數(shù)，真數(shù)大于0，。0log??xxya類型三、對(duì)數(shù)函數(shù)中的單調(diào)性問(wèn)題類型三、對(duì)數(shù)函數(shù)中的單調(diào)性問(wèn)題1函數(shù)的單調(diào)遞增區(qū)間為（）2()lg(43)fxxx???)1(??2函數(shù)的單調(diào)遞增區(qū)間是)152ln()(2???xxxf)5(??3函數(shù)的遞增區(qū)間是（）)23(log25.0???xxy)1(??4已知??312log981fxxx?????????，則??fx的最小值為（2）5若函數(shù)在區(qū)間上是增函數(shù)，的取值范圍。2

4、2log()yxaxa????(13)???a[2232]?6不等式的解集為1)12(log3??x]221(7設(shè)函數(shù)，且滿足，求的最大值。12.??????22log4log2fxxx??x241740xx?????fx提示：提示：（1）在對(duì)數(shù)函數(shù)中中，當(dāng)，在其定義域上是增函數(shù)；當(dāng)，在xxfalog)(?1?a)(xf01??a)(xf其定義域上是減函數(shù)。（2）在復(fù)合函數(shù)中，函數(shù)的單調(diào)性復(fù)合同增異減。)(log)(xgxfa?類型四、

5、對(duì)數(shù)函數(shù)中的大小比較類型四、對(duì)數(shù)函數(shù)中的大小比較1已知，比較，的大小。log4log4mn?mn01mn???2已知，比較的大小關(guān)系4log3log2log543???cbacbaabc??3設(shè)323loglog3log2abc????，則的大小關(guān)系abc??cba4若，，則B（A）（B）（C）（D）0??ba10??cccbaloglog?bbccloglog?ccba?bacc?5若，且，則與之間的大小關(guān)系是（）1?ayaxaaya

6、xloglog?????xy0??yx提示：提示：在比較大小題型中，當(dāng)，；當(dāng)，。byalog?1?a????????00101yxyx01??a????????00101yxyx類型五、對(duì)數(shù)函數(shù)求值問(wèn)題類型五、對(duì)數(shù)函數(shù)求值問(wèn)題1已知函數(shù)，若，則2xxflg)(?1)(?abf??)()(22bfaf2解方程或08log9loglog)(log32222????xx8?x41?x3已知若，，則，。1??ba25loglog??abbaab