高中數(shù)學(xué)人教高中必修第二章基本初等函數(shù)(Ⅰ)對(duì)數(shù)的概念王鑫

目錄CONTENTS感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺明晰對(duì)數(shù)之義FeelingthecomplexityoftheoperationExperiencingthemagicofnumbertableFindingthedefectofnumbertableClearingthedefinitionoflogarithm1324目錄CONTENTS感受運(yùn)算之繁Feelingthecomplexityoftheoperation體會(huì)數(shù)表之妙Experiencingthemagicofnumbertable發(fā)現(xiàn)數(shù)表之缺Findingthedefectofnumbertable明晰對(duì)數(shù)之義Clearingthedefinitionoflogarithm1324感受運(yùn)算之繁問題1:計(jì)算問題2:計(jì)算光在真空中的速度（千米/秒)一年的秒數(shù)1光年（天文單位）感受運(yùn)算之繁“沒有什么比大數(shù)的乘、除、開平方或開立方運(yùn)算更讓數(shù)學(xué)工作者頭痛、更阻礙計(jì)算者的了。這不僅浪費(fèi)時(shí)間，而且容易出錯(cuò)。因此，我開始考慮怎樣消除這些障礙……”困境:如何改進(jìn)數(shù)字計(jì)算方法？蘇格蘭數(shù)學(xué)家納皮爾（JohnNapier，1550-1617）目錄CONTENTS感受運(yùn)算之繁Feelingthecomplexityoftheoperation體會(huì)數(shù)表之妙Experiencingthemagicofnumbertable發(fā)現(xiàn)數(shù)表之缺Findingthedefectofnumbertable明晰對(duì)數(shù)之義Clearingthedefinitionoflogarithm1324感受運(yùn)算之繁體會(huì)數(shù)表之妙21222324252627209715241943048388608167772163355443267108864134217728x123456789102481632641282565121024111213141516171819202048409681921638432768655361310722621445242881048576282930313233…2684354565368709121073741824214748364842949672968589934592…問題3:計(jì)算感受運(yùn)算之繁體會(huì)數(shù)表之妙對(duì)數(shù)思想的來源之一：指數(shù)律降級(jí)思想：化乘、除為加、減，

化乘方、開方為乘、除感受運(yùn)算之繁體會(huì)數(shù)表之妙納皮爾將表中上行的數(shù)稱為下行中相應(yīng)數(shù)的

logarithm(對(duì)數(shù))x123456789102481632641282565121024logarithm/借數(shù)/對(duì)數(shù)真數(shù)康熙年間,數(shù)表傳入中國,《數(shù)理精蘊(yùn)》中把表中下行的數(shù)稱為“真數(shù)”,把上行的數(shù)稱為“借數(shù)”（也就是今天的“對(duì)數(shù)”,即對(duì)應(yīng)之?dāng)?shù)）“對(duì)數(shù)”名稱的由來目錄CONTENTS感受運(yùn)算之繁Feelingthecomplexityoftheoperation體會(huì)數(shù)表之妙Experiencingthemagicofnumbertable發(fā)現(xiàn)數(shù)表之缺Findingthedefectofnumbertable明晰對(duì)數(shù)之義Clearingthedefinitionoflogarithm1324感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺如何解決這個(gè)問題呢？第二行中相鄰兩項(xiàng)之間的間隔越來越大x1112131415161718204840968192163843276865536131072262144x2526272829335544326710886413421772826843545653687091231536299792458感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺運(yùn)動(dòng)模型納皮爾的對(duì)數(shù)方法（詳見教材P68頁閱讀材料“對(duì)數(shù)的發(fā)明”）感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺經(jīng)過20年堅(jiān)持不懈的研究，納皮爾出版了《奇妙的對(duì)數(shù)定律說明書》(1614)納皮爾的對(duì)數(shù)表感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺由指數(shù)函數(shù)可知，x存在且唯一.如何精確表示出其中的x？問題的本質(zhì):已知底數(shù)和冪的值,求指數(shù)x111213141516171820484096819216384327686553613107226214431536目錄CONTENTS感受運(yùn)算之繁Feelingthecomplexityoftheoperation體會(huì)數(shù)表之妙Experiencingthemagicofnumbertable發(fā)現(xiàn)數(shù)表之缺Findingthedefectofnumbertable明晰對(duì)數(shù)之義Clearingthedefinitionoflogarithm1324感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺對(duì)數(shù)的定義一般地，如果,那么，數(shù)x叫作以a為底N的對(duì)數(shù)記作：讀作：以a為底N的對(duì)數(shù)其中，a叫作對(duì)數(shù)的底數(shù)，N叫作真數(shù)明晰對(duì)數(shù)之義感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺明晰對(duì)數(shù)之義底數(shù)a的取值范圍：真數(shù)N的取值范圍：對(duì)數(shù)x的取值范圍：R指數(shù)式與對(duì)數(shù)式的互化感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺明晰對(duì)數(shù)之義兩個(gè)重要的恒等式：負(fù)數(shù)和零沒有對(duì)數(shù).感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺明晰對(duì)數(shù)之義例1.將下列指數(shù)式化為對(duì)數(shù)式，對(duì)數(shù)式化為指數(shù)式：例2.求下列各式中x的值：感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺明晰對(duì)數(shù)之義解：感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺明晰對(duì)數(shù)之義例3.（古巴比倫泥版上的復(fù)利問題）

年息20%，一定數(shù)目的錢經(jīng)過多長時(shí)間成為原來的兩倍?法國盧浮宮藏兩河流域泥版AO6770感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺明晰對(duì)數(shù)之義常用對(duì)數(shù)的誕生1615年，納皮爾與倫敦幾何學(xué)教授布里格斯的曠世之約納皮爾（J.Napier,1550-1617）布里格斯（H.Briggs，1561-1630）莫契斯頓城堡感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺明晰對(duì)數(shù)之義自然常數(shù)e與復(fù)利問題感受運(yùn)算之繁體會(huì)數(shù)表之妙發(fā)現(xiàn)數(shù)表之缺明晰對(duì)數(shù)之義德國數(shù)學(xué)家麥卡托(N.Mercator,

1620-1687)與自然對(duì)數(shù)其中e

≈2.71828…連續(xù)復(fù)利問題：年息100，一定數(shù)目的錢連續(xù)復(fù)利一年后，本利和為多少？對(duì)數(shù)的自白書或許大家都以為我源于指數(shù)，事實(shí)是我比指數(shù)的誕生早了幾步。君的迷茫，使我的誕生更具力量。我本只想躲在不易被人發(fā)現(xiàn)的地方，卻因?yàn)榧{皮爾，大家將我的價(jià)值比作太陽，

說我在簡化運(yùn)算方面光芒萬丈。

拉普拉斯說，“由于省時(shí)省力，我（對(duì)數(shù)）倍增了天文學(xué)家的壽命?！蹦苤茖W(xué)的發(fā)展，我是何等的榮幸。一直以來總有同學(xué)嘲笑我面目可