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积、商、幂的对数运算法则:积、商、幂的对数运算法则:如果如果a0,且,且a1,M0,N0有:有:NMMNaaaloglog)(logNMNMaaalogloglogR)M(nnManaloglogMnPMManPanpalogloglog)(logRnnananaaaaMMMMMlogloglog)M(log21n21MMaalog1log一、对数的换底公式一、对数的换底公式:如何证明呢如何证明呢?aNNccalogloglog)0),1()1,0(,(Nca证明证明:设:设 由对数的定义可以得:由对数的定义可以得:paN 即证得即证得 pNalogpccaNloglogapNccloglogaNpccloglogaNNccalogloglog通过换底公式,人们通过换底公式,人们可以把其他底的对数可以把其他底的对数转换为以转换为以10或或e为底为底的对数,经过查表就的对数,经过查表就能求出任意不为能求出任意不为1的的正数为底的对数。正数为底的对数。二、几个重要的推论二、几个重要的推论:如何证明呢如何证明呢?abbalog1logNmnNanamloglog),1()1,0(,ba证明证明:利用换底公式得:利用换底公式得:即证得即证得 NmnNanamlogloglglglgloglglglgmnaNnNnNnNamamamlogaNmnaNlglg证明证明:由换底公式由换底公式 abbalog1log即即 abbaloglog1lglglglgbaab1logloglogacbcba推论推论:例例1:计算计算:解解:27log19 27log19333log23log23323 8log7log3log2732 9lg212log1100333 9lg212log1100333 8log7log3log27322lg2lg32lg3lg3lg7lg7lg8lg3解解:例例1:计算计算:27log19 8log7log3log2732解解:9lg212log11003339lg2122log103339lg102392315 9lg212log1100333例例1:计算计算:27log19 8log7log3log2732解解:.)21(2,10054:2的值求设例baba10054ba10log10log100log22242a2log224log245log100log55255b2log1110log12)21(252ba25log2log22log5log12log210105510.9log,7log,5log:33539表示试用已知例nmnm解解:7log,5log215log5log33392nm7log,25log33nmnm227log5log235log23log29log3333535.,07lg5lglg)7lg5(lglg:421212xxxxxx求的两根分别为方程例07lg5lglg)7lg5(lglg2xx解解:7lg5lglglg)7lg5(lglglg2121xxxx351lg35lg35lglg121xx35121xx
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