对数与对数函数(课堂PPT).ppt

单击此处编辑母版标题样式,编辑母版文本样式,第二级,第三级,第四级,第五级,*,第七节 对数与对数函数,1,1.,对数概念,(1),定义,:,一般地,如果,那么,x,叫,做,记作,其中,a,叫做对数的,N,叫做,.(2),对数性质,没有对数,即,;1,的对数为,0,即,;,底的对数等于,1,即,.,基础梳理,真数,=,N,(,a,0,，且,a,1),以,a,为底,N,的对数,底数,零和负数,N,0,log,a,1=0(,a,0,，且,a,1),log,a,a,=1(,a,0,，且,a,1),2,(3),对数恒等式,:.,(4),常用对数,:,通常将,叫做常用对数,N,的常用对数,log,10,N,简记为,.(5),自然对数,:,称为自然对数,N,的自然对数,log,e,N,简记作,.,a,log,a,N,=,N,(,a,0,，且,a,1,，,N,0),以,10,为底的对数,lg,N,以无理数,e=2.718 28,为底的对数,ln,N,3,2.,对数的运算性质如果,a,0,且,a,1,M0,N0,那么,(1),=log,a,M+log,a,N;(2)log,a,=,;,(3),=nlog,a,M(nR).,log,a,M,n,log,a,(,M,N,),log,a,M,-log,a,N,4,3.,换底公式及常见结论,(1),换底公式,:,(a,b0,且,a,b 1,N0),.(2)常见结论,(,其中,a,b,c0,且,a,b,c1):log,a,=,log,a,b=,-1,log,a,b,1,5,4.,对数函数的定义,:,一般地,函数,叫做对数函数,它的定义域为,值域为,.5.,对数函数的图象与性质,y,=log,a,x,(,a,0,，,a,1),R,(0，+),6,y,=log,a,x,a,1,0,a,1,时，,_,；,当,0,x,1,时，,_,；,当,0,x,0,y,0,y,0,增函数,减函数,8,6.,反函数,指数函数,y,=,a,x,(,a,0,，,a,1),与对数函数,y,=log,a,x,(,a,0,，,a,1,，,x,0),，,它们的图象关于直线,_,对称,y,=,x,互为反函数,9,基础达标,(,教材改编题,),若,a,0,，,a,1,，,x,y,0,，,n,N,*,，,则下列各式正确的有,(,),(log,a,x,),n,=,n,log,a,x,；,(log,a,x,),n,=log,a,x,n,；,log,a,x,=,log,a,；,log,a,x,=log,a,n,x,n,log,a,(,x,y,)=log,a,x,log,a,y,；,log,a,(,xy,)=log,a,x,log,a,y,.,A.4,个,B.5,个,C.6,个,D.7,个,10,2.(,教材改编题,),对于,a,0,，,a,1,，下列说法正确,的是,(,),若,MN,，则,log,a,M,=log,a,N,；,若,log,a,M,=log,a,N,，则,M,=,N,；,若,log,a,M,2,=log,a,N,2,，则,M,=,N,；,若,M,=,N,，,log,a,M,2,=log,a,N,2,.,A.,B.,C.,D.,1.A,解析：正确，故选,A,C,解析：,错，因为,M,=,N,0,时，对数无意义；正确；,错，因为,log,a,M,2,=log,a,N,2,，则,|,M,|=|,N,|,；,错，若,M,=,N,=0,，则对数无意义，故选,C.,11,3.,若,y,=log,5,6log,6,7log,7,8log,8,9log,9,10,则有,(,),A.,y,(0,1),B.,y,(1,2),C.,y,(2,3),D.,y,1,B,解析：,log,5,5,log,5,10,log,5,5,2,，,1,log,5,10,2,，,y,(1,2),12,4.(1-log,6,3),2,+log,6,2log,6,18,log,6,4=_.,解析：原式,=(log,6,2),2,+log,6,2(1+log,6,3),2log,6,2,=(log,6,2),2,+log,6,2+log,6,2,log,6,3,2log,6,2,=,log,6,2+,+,log,6,3,log,6,(23)+,=,+,=1.,=,1,13,经典例题题型一对数的运算,【,例,1】,求下列各式的值,(1),已知,lg,x,+lg,y,=2lg(,x,-2,y,),，求的值；,(2)(2010,辽宁改编,),设,2,a,=5,b,=,m,，且,求,m,的值,14,解：,(1),由题意可得,x,0,，,y,0,，且,x,2,y,.,又,lg,x,+lg,y,=2lg(,x,-2,y,),，,xy,=(,x,-2,y,),2,，即,x,2,-5,xy,+4,y,2,=0,，,解得,x,=4,y,(,或,x,=,y,舍去,),(2)2,a,=5,b,=,m,，,a,=log,2,m,，,b,=log,5,m,，,m,2,=10,，,m,0,，,m15,【,例,2】,(2010,天津,),设,a,=log,5,4,，,b,=(log,5,3),2,，,c,=log,4,5,，则,(,),A.,a,c,b,B.,b,c,a,C.,a,b,c,D.,b,a,c,题型二对数值,(,式,),的大小比较,16,解：,a,=log,5,4,1,0,log,5,3,log,5,4,1,，,b,=(log,5,3),2,log,5,3,log,5,4=,a,.,又,c,=log,4,5,1,，,b,a,c,.,故选,D.,17,变式,2-1,比较下列各数的大小：,1.1,0.9,，,log,1.1,0.9,，,log,0.7,0.8.,解析：,log,1.1,0.9,log,1.1,1=0,，,log,0.7,0.8,log,0.7,1=0,1.1,0.9,1.1,0,=1,，,又,log,0.7,0.8,log,0.7,0.7=1,，,1.1,0.9,log,0.7,0.8,log,1.1,0.9.,18,【,例,3】,求函数,y,=log,a,(,x,-,x,2,)(,a,1),的定义域、,值域、单调区间,题型三对数函数的图象与性质的应用,19,.,解：由,x,-,x,2,0,得,0,x,1,，,所以函数,y,=log,a,(,x,-,x,2,),的定义域是,(0,1),因为,0,x,-,x,2,=,所以，当,a,1,时，,log,a,(,x,-,x,2,)log,a,函数,y,=log,a,(,x,-,x,2,),的值域为,令,u,=,x,-,x,2,(0,x,1),，,因为,u,=,x,-,x,2,在 上是增函数，在 上是减函数，,由对数函数,y,=log,a,u,的单调性，可得：,当,a,1,时，函数,y,=log,a,(,x,-,x,2,),在 上是增函数,上是减函数,故函数的单调递减区间 是递增区间是,20,(2011,雅礼中学月考,),若集合,A,=,x,|log,a,(,x,2,-,x,-2),2,，,a,0,且,a,1,(1),若,a,=2,，求集合,A,；,(2),若 ,A,，求实数,a,的取值范围变式,3-1,解析：,(1),若,a,=2,，,log,2,(,x,2,-,x,-2),2,，则,x,2,-,x,-2,4,，,解得,x,-2,或,x,3.,所以,A,=,x,|,x,-2,或,x,3,21,(2),因为 ，所以,当,a,1,时，无解；,当,0,a,1,时，解得,22,易错警示,若函数,y,=log,2,的值域为,R,，求实数,a,的取值范围,错解函数,y,=log,2,的值域为,R,，,对任意实数,ax,2,+(,a,-1),x,+,恒成立,a,=0,时不成立,若,a,0,时，则 即,解得,23,错解分析函数,y,=log,2,的值域为,R,，,(0,，,+),必须是,u,=,ax,2,+(,a,-1),x,+,值域的子集,也就是函数,u,=,ax,2,+(,a,-1),x,+,必须开口向上且与,x,轴有交点,24,解得,0,a,或,a,当,a,=0,时，,u,(,x,)=,x,+,满足题意,a,的取值范围是,0,a,或,a,25,