高中数学对数
\u9ad8\u4e2d\u6570\u5b66\u5bf9\u6570\u8fd0\u7b971.\u4e24\u8fb9\u53d6\u5bf9\u6570\uff0c\u4ee5\u6839\u53f72\u51cf1\u4e3a\u5e95\u6570
x=log(\u6839\u53f72\u51cf1\uff092
2.\u53ef\u77e5\uff1alog2 x+log2 y=log2 (xy)=2
\u5219xy=4\uff0c\u56e0\u4e3ax\uff0cy\u22650
\u800c\uff08x+y)\u22652\u6839\u53f7\uff08xy\uff09=4
3.1+1/2lg9-lg240=1+lg3-lg\uff0816*15\uff09
=1+lg3-lg16-lg15
=1+lg3-4lg2-lg3-lg5
=lg10-4lg2-lg5
=lg2+lg5-4lg2-lg5
=-3lg2
1-2/3lg27+lg36/5=1-2/3lg3^3+lg36-lg5
=1-2lg3+lg36-lg5
=1-2lg3+2lg6-lg5
=1-2lg3+2lg2+2lg3-lg5
=lg10+2lg2-lg5
=lg2+lg5+2lg2-lg5
=3lg2
\u5219\u5206\u5b501+1/2lg9-lg240\u9664\u4ee5\u5206\u6bcd1-2/3lg27+lg36/5\u7b49\u4e8e\uff1a-1
4.lg4+lg5lg20+(lg5)^2
=2lg2+lg5*\uff08lg5+lg20\uff09
=2lg2+lg5*lg100
=2lg2+2lg5
=2lg10
=2
5.log\u4ee53\u4e3a\u5e95\u76844\u4e58\u4ee5log\u4ee54\u4e3a\u5e95\u76845\u4e58\u4ee5log\u4ee55\u4e3a\u5e95\u76846\u4e00\u76f4\u4e58\u5230log\u4ee580\u4e3a\u5e95\u768481
\u6362\u4e3a\uff1a \u4ee5\u81ea\u7136\u5bf9\u6570\u4e3a\u5e95\u7684\u5bf9\u6570\u3002
=lg4/lg3*lg5/lg4*...lg81/80
\u4e2d\u95f4\u7684\u5168\u90e8\u6d88\u9664
=lg81/lg3
=lg3^4/lg3
=4lg3/lg3
=4
6.log\u4ee542\u4e3a\u5e95\u768456=log\u4ee542\u4e3a\u5e95\u76847*8
\u6362\u5e95\u53ef\u5f97\uff1a
=lg56/lg42
=lg\uff087*8\uff09/lg(6*7)
=\uff08lg7+3lg2\uff09/\uff08lg2+lg3+lg7\uff09
\u53c8log\u4ee52\u4e3a\u5e95\u76843\u7b49\u4e8ea\uff0clog\u4ee53\u4e3a\u5e95\u76847\u7b49\u4e8eb
\u6362\u5e95\u53ef\u5f97
lg3/lg2=a\uff0clg7/lg3=b
\u6240\u4ee5lg7=blg3
lg2=1/alg3
\u4ee3\u5165\u5f97\uff1a
\uff08lg7+3lg2\uff09/\uff08lg2+lg3+lg7\uff09
=\uff08blg3+3/alg3\uff09/\uff081/alg3+lg3+blg3\uff09
=\uff08b+3/a\uff09/(1/a+1+b)
=\uff08ab+3)/(ab+a+1)
y=f(x)=log.2.x,\u52192\uff3ey=x\uff0c\u2234g(x)=2\uff3ex
g(a)g(b)=(2\uff3ea)*(2\uff3eb)=2\uff3e(a+b)=8,\u2234a+b=3
\u2235a>0,b>0,\u22340<a<3,b=3-a
u=1/a+4/b=1/a+4/(3-a),u'=-1/a\uff3e2+4/(3-a)\uff3e2
\u4ee4u'=0,\u5f972a=\u00b1(a-3),\u22350<a<3,\u2234a=1,\u800cb=2
\u5bb9\u6613\u77e5\u9053\uff0cu\u57280<a<3\u65f6\uff0c\u6700\u5c0f\u70b9\u662fa=1\uff0c\u5176\u6700\u5c0f\u503c\u662f1+2=3
推倒一：
设a^b=N…………①
则b=logaN…………②
把②代入①即得对数恒等式：
a^(logaN)=N…………③
把③两边取以m为底的对数得
logaN·logma=logmN
所以
logaN=(logmN)/(logma)
推导2：
设t=log(a)b
则有a^t=b
两边取以e为底的对数
tlna=lnb
t=lnb/lna
即是：log(a)b=lnb/lna
求证：logaN=logmN
-----
logma
证明：由logaN=b可得a的b次方=N
两边取m(m>0且m≠1)为底的对数，得blogma=logmN,
∵a≠1,∴logma≠0,∴b=logmN ,∴logaN=logmN
----- -----
logma logma
输入时角标有些困难，凑合看吧
绛旓細log鍦楂樹腑鏁板閲岃〃绀瀵规暟銆備竴鑸湴锛屽嚱鏁皔=logax锛坅>0锛屼笖a鈮1锛夊彨鍋氬鏁板嚱鏁帮紝涔熷氨鏄浠ュ箓锛堢湡鏁帮級涓鸿嚜鍙橀噺锛屾寚鏁颁负鍥犲彉閲忥紝搴曟暟涓哄父閲忕殑鍑芥暟锛屽彨瀵规暟鍑芥暟銆傞氬父鎴戜滑灏嗕互10涓哄簳鐨勫鏁板彨甯哥敤瀵规暟锛坈ommon logarithm锛夛紝骞舵妸log10N璁颁负lgN銆傚彟澶栵紝鍦ㄧ瀛﹁鏁颁腑甯镐娇鐢ㄤ互鏃犵悊鏁癳=2.71828路路路涓哄簳鏁...
绛旓細楂樹腑鏁板瀵规暟鍏紡澶у叏濡備笅锛1銆佸鏁拌繍绠楁硶鍒欙細a^log锛坅锛塏=N锛坅>0涓攁涓嶇瓑浜1)锛塴og锛坅锛塣n=n锛坅>0涓攁涓嶇瓑浜1锛塴og锛坅锛塎N=log锛坅锛塎+log锛坅锛塏锛坅>0鏈坅涓嶇瓑浜1锛夈俵og锛坅锛塎/N=log锛坅锛塎-log锛坅锛塏锛坅>0鏈坅涓嶇瓑浜1锛夈俵og锛坅锛塣M^n=nlog锛坅锛塣M锛坅>0鏈坅涓嶇瓑浜1锛夈...
绛旓細鍦楂樹腑鏁板涓紝log锛瀵规暟锛夋槸鎸囨暟涓庡鏁颁箣闂寸殑鏁板鍏崇郴銆傚鏁版槸鎸囦竴涓暟锛堣绉颁负鐪熸暟锛夊湪鏌愪釜鍩烘暟涓嬬殑鎸囨暟锛屽彲浠ヨ〃绀轰负浠ヤ笅褰㈠紡锛歭ogₐ(x) = y 鍏朵腑锛宎 鏄熀鏁帮紙涓鑸负姝ｅ疄鏁颁笖涓嶇瓑浜1锛夛紝x 鏄湡鏁帮紙姝ｅ疄鏁帮級锛寉 鏄寚鏁般傚鏁扮殑瀹氫箟鏉ユ簮浜庢寚鏁拌繍绠楃殑閫嗚繍绠椼傞氳繃姹傝В瀵规暟锛屾垜浠彲浠ュ緱鍒版寚鏁...
绛旓細楂樹腑鏁板涓璴og鐭ヨ瘑鐐瑰涓嬶細1銆瀵规暟鍏紡鏄暟瀛︿腑鐨勪竴绉嶅父瑙佸叕寮忥紝濡傛灉a^x=N(a>0,涓攁鈮1)锛屽垯x鍙仛浠涓哄簳N鐨勫鏁帮紝璁板仛x=log(a)(N)锛屽叾涓璦瑕佸啓浜巐og鍙充笅銆傚叾涓璦鍙仛瀵规暟鐨勫簳锛孨鍙仛鐪熸暟銆2銆侀氬父鎴戜滑灏嗕互10涓哄簳鐨勫鏁板彨鍋氬父鐢ㄥ鏁帮紝浠涓哄簳鐨勫鏁扮О涓鸿嚜鐒跺鏁般3銆佸鏁扮殑鍏紡閮芥湁loga(1...
绛旓細log鍦楂樹腑鏁板閲岃〃绀瀵规暟銆傚鏋渁^n = b锛坅>0,涓攁鈮1锛,閭ｄ箞鏁皀鍙仛浠涓哄簳b鐨勫鏁,璁板仛n=log(a)b,銆恆鏄笅鏍囥戝叾涓,a鍙仛鈥滃簳鏁扳,b鍙仛鈥滅湡鏁扳濄備竴鑸湴锛屽嚱鏁皔=logax锛坅>0锛屼笖a鈮1锛夊彨鍋氬鏁板嚱鏁帮紝涔熷氨鏄浠ュ箓锛堢湡鏁帮級涓鸿嚜鍙橀噺锛屾寚鏁颁负鍥犲彉閲忥紝搴曟暟涓哄父閲忕殑鍑芥暟锛屽彨瀵规暟鍑芥暟銆傚叾涓...
绛旓細鍦楂樹腑鏁板涓紝瀵规暟鏄竴涓潪甯搁噸瑕佺殑姒傚康锛屽畠鍦ㄨВ鍐冲疄闄呴棶棰樺拰鎺ㄥ鏁板鍏紡鏃堕兘鏈夌潃骞挎硾鐨勫簲鐢ㄣ備互涓嬫槸楂樹腑鐢熼渶瑕佹帉鎻＄殑瀵规暟鐭ヨ瘑锛1.瀵规暟鐨勫畾涔夛細瀵规暟鏄涓涓暟杩涜鎸囨暟杩愮畻鐨勭粨鏋溿備緥濡傦紝log_ab琛ㄧず浠涓哄簳b鐨勫鏁般2.瀵规暟鐨勬ц川锛氬鏁板叿鏈変竴浜涘熀鏈殑鎬ц川锛屽鎹㈠簳鍏紡銆佸鏁扮殑鍔犲噺娉曟ц川銆佸鏁扮殑涔橀櫎...
绛旓細1瀵规暟鐨勬蹇 濡傛灉a(a>0,涓攁鈮1)鐨刡娆″箓绛変簬N,鍗砤b=N,閭ｄ箞鏁癰鍙仛浠涓哄簳N鐨勫鏁,璁颁綔锛歭ogaN=b,鍏朵腑a鍙仛瀵规暟鐨勫簳鏁,N鍙仛鐪熸暟.鐢卞畾涔夌煡锛氣憼璐熸暟鍜岄浂娌℃湁瀵规暟; 鈶>0涓攁鈮1,N>0; 鈶oga1=0,logaa=1,alogaN=N,logaab=b.鐗瑰埆鍦,浠10涓哄簳鐨勫鏁板彨甯哥敤瀵规暟,璁颁綔log10N,绠璁颁负lgN...
绛旓細楂樹腑鏁板涓殑lg鍏紡鏄寚浠10涓哄簳鐨瀵规暟鍑芥暟銆傚叾琛ㄧず褰㈠紡涓猴細lg(x)=log10(x)銆備笅闈㈠垪涓句竴浜涘父瑙佺殑lg鍏紡鍙婂叾鎬ц川锛1.lg(1)=0锛氫换浣曟暟鐨勫鏁颁互10涓哄簳鏃讹紝瀵瑰簲鐨勫间负0銆2.lg(10)=1锛10鐨勫鏁颁互10涓哄簳鏃讹紝瀵瑰簲鐨勫间负1銆3.瀵规暟鐨勪箻绉硶鍒欙細lg(a*b)=lg(a)+lg(b)4.瀵规暟鐨勫晢娉曞垯锛歭g(a...
绛旓細甯哥敤瀵规暟涓庤嚜鐒跺鏁帮細甯哥敤瀵规暟浠10涓哄簳锛岃浣 lg锛堟垨鏈夋椂鐪佺暐 lg锛夛紝鑰岃嚜鐒跺鏁颁互鏁板甯告暟e锛堢害绛変簬2.718锛変负搴曪紝璁颁綔 ln銆傚鑷劧瀵规暟鐨勮璇嗛渶瑕佷竴瀹氱殑楂樼瓑鏁板鐭ヨ瘑锛楂樹腑鐢熷彧闇浜嗚В鍏跺熀鏈蹇点傚鏁拌繍绠楄鍒欙細鎺屾彙浜嗚繖浜涜鍒欙紝濡 log_b(xy) = log_b x + log_b y锛宭og_b(x^y) = y log...
绛旓細鍦楂樹腑鏁板涓紝瀵规暟锛坙ogarithm锛夋槸涓涓噸瑕佺殑姒傚康銆備笅闈㈡垜灏嗘彁渚涗竴涓缁嗙殑瑙ｆ瀽锛屼粙缁嶅浣曡绠楀鏁般傚鏁版槸鎸囨暟杩愮畻鐨勯嗚繍绠椼傞氬父鎯呭喌涓嬶紝鎴戜滑浣跨敤浠10涓哄簳鐨勫父鐢ㄥ鏁帮紙log锛夛紝浠ュ強浠ヨ嚜鐒跺父鏁癳涓哄簳鐨勮嚜鐒跺鏁帮紙ln锛夈傚浜庢櫘閫氬鏁般佽嚜鐒跺鏁颁互鍙婂叾浠栧簳鏁扮殑瀵规暟璁＄畻鏂规硶鍩烘湰鐩稿悓锛屽彧鏄簳鏁颁笉鍚岃屽凡銆傛櫘閫氬鏁...
