高中数学对数计算
\u9ad8\u4e2d\u6570\u5b66\u7684\u6240\u6709\u5bf9\u6570\u8ba1\u7b97\u516c\u5f0f \u6025\u554a\u5b9a\u4e49\uff1a
\u3000\u3000\u82e5a^n=b(a>0\u4e14a\u22601)
\u3000\u3000\u5219n=log(a)(b)
\u3000\u3000\u57fa\u672c\u6027\u8d28\uff1a
\u3000\u30001\u3001a^(log(a)(b))=b
\u3000\u30002\u3001log(a)(MN)=log(a)(M)+log(a)(N);
\u3000\u30003\u3001log(a)(M\u00f7N)=log(a)(M)-log(a)(N);
\u3000\u30004\u3001log(a)(M^n)=nlog(a)(M)
\u3000\u3000\u63a8\u5bfc
\u3000\u30001\u3001\u56e0\u4e3an=log(a)(b)\uff0c\u4ee3\u5165\u5219a^n=b\uff0c\u5373a^(log(a)(b))=b\u3002
\u3000\u30002\u3001MN=M\u00d7N
\u3000\u3000\u7531\u57fa\u672c\u6027\u8d281(\u6362\u6389M\u548cN)
\u3000\u3000a^[log(a)(MN)] = a^[log(a)(M)]\u00d7a^[log(a)(N)]
\u3000\u3000\u7531\u6307\u6570\u7684\u6027\u8d28
\u3000\u3000a^[log(a)(MN)] = a^{[log(a)(M)] + [log(a)(N)]}
\u3000\u3000\u53c8\u56e0\u4e3a\u6307\u6570\u51fd\u6570\u662f\u5355\u8c03\u51fd\u6570\uff0c\u6240\u4ee5
\u3000\u3000log(a)(MN) = log(a)(M) + log(a)(N)
\u3000\u30003\u3001\u4e0e\uff082\uff09\u7c7b\u4f3c\u5904\u7406
\u3000\u3000MN=M\u00f7N
\u3000\u3000\u7531\u57fa\u672c\u6027\u8d281(\u6362\u6389M\u548cN)
\u3000\u3000a^[log(a)(M\u00f7N)] = a^[log(a)(M)]\u00f7a^[log(a)(N)]
\u3000\u3000\u7531\u6307\u6570\u7684\u6027\u8d28
\u3000\u3000a^[log(a)(M\u00f7N)] = a^{[log(a)(M)] - [log(a)(N)]}
\u3000\u3000\u53c8\u56e0\u4e3a\u6307\u6570\u51fd\u6570\u662f\u5355\u8c03\u51fd\u6570\uff0c\u6240\u4ee5
\u3000\u3000log(a)(M\u00f7N) = log(a)(M) - log(a)(N)
\u3000\u30004\u3001\u4e0e\uff082\uff09\u7c7b\u4f3c\u5904\u7406
\u3000\u3000M^n=M^n
\u3000\u3000\u7531\u57fa\u672c\u6027\u8d281(\u6362\u6389M)
\u3000\u3000a^[log(a)(M^n)] = {a^[log(a)(M)]}^n
\u3000\u3000\u7531\u6307\u6570\u7684\u6027\u8d28
\u3000\u3000a^[log(a)(M^n)] = a^{[log(a)(M)]*n}
\u3000\u3000\u53c8\u56e0\u4e3a\u6307\u6570\u51fd\u6570\u662f\u5355\u8c03\u51fd\u6570\uff0c\u6240\u4ee5
\u3000\u3000log(a)(M^n)=nlog(a)(M)
\u3000\u3000\u57fa\u672c\u6027\u8d284\u63a8\u5e7f
\u3000\u3000log(a^n)(b^m)=m/n*[log(a)(b)]
\u3000\u3000\u63a8\u5bfc\u5982\u4e0b\uff1a
\u3000\u3000\u7531\u6362\u5e95\u516c\u5f0f\uff08\u6362\u5e95\u516c\u5f0f\u89c1\u4e0b\u9762\uff09[lnx\u662flog(e)(x)\uff0ce\u79f0\u4f5c\u81ea\u7136\u5bf9\u6570\u7684\u5e95]
\u3000\u3000log(a^n)(b^m)=ln(b^m)\u00f7ln(a^n)
\u3000\u3000\u7531\u57fa\u672c\u6027\u8d284\u53ef\u5f97
\u3000\u3000log(a^n)(b^m) = [m\u00d7ln(b)]\u00f7[n\u00d7ln(a)] = (m\u00f7n)\u00d7{[ln(b)]\u00f7[ln(a)]}
\u3000\u3000\u518d\u7531\u6362\u5e95\u516c\u5f0f
\u3000\u3000log(a^n)(b^m)=m\u00f7n\u00d7[log(a)(b)] --------------------------------------------\uff08\u6027\u8d28\u53ca\u63a8\u5bfc \u5b8c\uff09
\u7f16\u8f91\u672c\u6bb5\u51fd\u6570\u56fe\u8c61
\u3000\u30001.\u5bf9\u6570\u51fd\u6570\u7684\u56fe\u8c61\u90fd\u8fc7(1,0)\u70b9.
\u3000\u30002.\u5bf9\u4e8ey=log(a)(n)\u51fd\u6570,
\u3000\u3000\u2460,\u5f530<a<1\u65f6,\u56fe\u8c61\u4e0a\u51fd\u6570\u663e\u793a\u4e3a(0,+\u221e)\u5355\u51cf.\u968f\u7740a \u7684\u589e\u5927,\u56fe\u8c61\u9010\u6e10\u4ee5(1,0)\u70b9\u4e3a\u8f74\u987a\u65f6\u9488\u8f6c\u52a8,\u4f46\u4e0d\u8d85\u8fc7X=1.
\u3000\u3000\u2461\u5f53a>1\u65f6,\u56fe\u8c61\u4e0a\u663e\u793a\u51fd\u6570\u4e3a(0,+\u221e)\u5355\u589e,\u968f\u7740a\u7684\u589e\u5927,\u56fe\u8c61\u9010\u6e10\u4ee5(1.0)\u70b9\u4e3a\u8f74\u9006\u65f6\u9488\u8f6c\u52a8,\u4f46\u4e0d\u8d85\u8fc7X=1.
\u3000\u30003.\u4e0e\u5176\u4ed6\u51fd\u6570\u4e0e\u53cd\u51fd\u6570\u4e4b\u95f4\u56fe\u8c61\u5173\u7cfb\u76f8\u540c,\u5bf9\u6570\u51fd\u6570\u548c\u6307\u6570\u51fd\u6570\u7684\u56fe\u8c61\u5173\u4e8e\u76f4\u7ebfy=x\u5bf9\u79f0.
\u7f16\u8f91\u672c\u6bb5\u5176\u4ed6\u6027\u8d28
\u3000\u3000\u6027\u8d28\u4e00\uff1a\u6362\u5e95\u516c\u5f0f
\u3000\u3000log(a)(N)=log(b)(N)\u00f7log(b)(a)
\u3000\u3000\u63a8\u5bfc\u5982\u4e0b\uff1a
\u3000\u3000N = a^[log(a)(N)]
\u3000\u3000a = b^[log(b)(a)]
\u3000\u3000\u7efc\u5408\u4e24\u5f0f\u53ef\u5f97
\u3000\u3000N = {b^[log(b)(a)]}^[log(a)(N)] = b^{[log(a)(N)]*[log(b)(a)]}
\u3000\u3000\u53c8\u56e0\u4e3aN=b^[log(b)(N)]
\u3000\u3000\u6240\u4ee5 b^[log(b)(N)] = b^{[log(a)(N)]*[log(b)(a)]}
\u3000\u3000\u6240\u4ee5 log(b)(N) = [log(a)(N)]*[log(b)(a)] {\u8fd9\u6b65\u4e0d\u660e\u767d\u6216\u6709\u7591\u95ee\u770b\u4e0a\u9762\u7684}
\u3000\u3000\u6240\u4ee5log(a)(N)=log(b)(N) / log(b)(a)
\u3000\u3000\u516c\u5f0f\u4e8c\uff1alog(a)(b)=1/log(b)(a)
\u3000\u3000\u8bc1\u660e\u5982\u4e0b\uff1a
\u3000\u3000\u7531\u6362\u5e95\u516c\u5f0f log(a)(b)=log(b)(b)/log(b)(a) ----\u53d6\u4ee5b\u4e3a\u5e95\u7684\u5bf9\u6570
\u3000\u3000log(b)(b)=1 =1/log(b)(a) \u8fd8\u53ef\u53d8\u5f62\u5f97: log(a)(b)\u00d7log(b)(a)=1
\u3000\u3000\u5728\u5b9e\u7528\u4e0a\uff0c\u5e38\u91c7\u7528\u4ee510\u4e3a\u5e95\u7684\u5bf9\u6570\uff0c\u5e76\u5c06\u5bf9\u6570\u8bb0\u53f7\u7b80\u5199\u4e3algb,\u79f0\u4e3a\u5e38\u7528\u5bf9\u6570\uff0c\u5b83\u9002\u7528\u4e8e\u6c42\u5341\u8fdb\u4f2f\u5236\u6574\u6570\u6216\u5c0f\u6570\u7684\u5bf9\u6570\u3002\u4f8b\u5982lg10=1,lg100=lg102=2,lg4000=lg\uff08103\u00d74\uff09=3+lg4,\u53ef\u89c1\u53ea\u8981\u5bf9\u67d0\u4e00\u8303\u56f4\u7684\u6570\u7f16\u5236\u51fa\u5bf9\u6570\u8868\uff0c\u4fbf\u53ef\u5229\u7528\u6765\u8ba1\u7b97\u5176\u4ed6\u5341\u8fdb\u5236\u6570\u7684\u5bf9\u6570\u7684\u8fd1\u4f3c\u503c\u3002\u5728\u6570\u5b66\u7406\u8bba\u4e0a\u4e00\u822c\u90fd\u7528\u4ee5\u65e0\u7406\u6570e=2.7182818\u2026\u2026\u4e3a\u5e95\u7684\u5bf9\u6570\uff0c\u5e76\u5c06\u8bb0\u53f7 loge\u3002\u7b80\u5199\u4e3aln\uff0c\u79f0\u4e3a\u81ea\u7136\u5bf9\u6570\uff0c\u56e0\u4e3a\u81ea\u7136\u5bf9\u6570\u51fd\u6570\u7684\u5bfc\u6570\u8868\u8fbe\u5f0f\u7279\u522b\u7b80\u6d01\uff0c\u6240\u4ee5\u663e\u51fa\u4e86\u5b83\u6bd4\u5176\u4ed6\u5bf9\u6570\u5728\u7406\u8bba\u4e0a\u7684\u4f18\u8d8a\u6027\u3002\u5386\u53f2\u4e0a\uff0c\u6570\u5b66\u5de5\u4f5c\u8005\u4eec\u7f16\u5236\u4e86\u591a\u79cd\u4e0d\u540c\u7cbe\u786e\u5ea6\u7684\u5e38\u7528\u5bf9\u6570\u8868\u548c\u81ea\u7136\u5bf9\u6570\u8868\u3002\u4f46\u968f\u7740\u7535\u5b50\u6280\u672f\u7684\u53d1\u5c55\uff0c\u8fd9\u4e9b\u6570\u8868\u5df2\u9010\u6e10\u88ab\u73b0\u4ee3\u7684\u7535\u5b50\u8ba1\u7b97\u5de5\u5177\u6240\u53d6\u4ee3\u3002
\u5de6\u8fb9lga+lgb=lgab
\u53f3\u8fb92lg(a-2b)=lg(a-2b)2
\u5de6\u8fb9\uff1d\u53f3\u8fb9\uff0c
\u5219ab=(a-2b)2
\u4e24\u8fb9\u9664\u4ee5b2
\u5316\u7b80\u4e3a:a/b=(a/b-2)2
\u53e6a/b=x
\u5373x=(x-2)2
x2-5x+4=0;
(x-1)(x-4)=0
x=1\u6216x=4
\u5f53x=1\u65f6\uff0ca-2b=-b<0;\u4e0d\u7b26\u5408\u6761\u4ef6
\u6545a/b=4
\u671b\u91c7\u7eb3\uff0c\u8c22\u8c22\uff0c\u6709\u4e0d\u61c2\u7684\u53ef\u4ee5\u7ee7\u7eed\u95ee\u6211
运用对数的换底公式
（log2 125+log4 25+log8 5）*（log125 8+log25 4+log5 2）
=(3*log2 5+log2 5+1/3*log2 5)*(log5 2+log5 2+log5 2)
=13/3*log2 5*3*log5 2
=13
原式=(log2 125+1/2log2 25+1/3log2 5)*(4/log2 125+2/log2 25+1/log2 5)
=(3log2 5+log2 5+1/4log2 5)*(log2 5+1/log2 5+1/log2 5)
=((13/3)*log2 5)*(3*(1/log2 5))
=3
（log2 125+log4 25+log8 5）*（log125 8+log25 4+log5 2）=（log2 125+1/2log2 25+1/3log2 5）*（1/3log5 8+1/2log5 4+log5 2）=（log2 125+log2 5+log2 5^1/3）*（log5 2+log5 2+log5 2）=log2 5^13/3*log5 2^3=13/3*3*log2 5*log5 2=13
绛旓細楂樹腑鏁板瀵规暟鍏紡澶у叏濡備笅锛1銆瀵规暟杩愮畻娉曞垯锛歛^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...
绛旓細瀵规暟鏄珮涓鏁板蹇呬慨涓瀛︾殑銆傚鏁扮殑杩愮畻娉曞垯锛1銆乴og(a) (M路N锛=log(a) M+log(a) N 2銆乴og(a) (M梅N)=log(a) M-log(a) N 3銆乴og(a) M^n=nlog(a) M 4銆乴og(a)b*log(b)a=1 5銆乴og(a) b=log (c) b梅log (c) a 瀵规暟搴旂敤 瀵规暟鍦ㄦ暟瀛﹀唴澶栨湁璁稿搴旂敤銆傝繖浜涗簨浠朵腑鐨...
绛旓細1銆侀鍏堬紝鎷垮嚭鍗¤タ娆璁＄畻棰橈紝鎸塷n閿紑鏈恒2銆佺浜屾鍦ㄨ緭鍏ユ爮杈撳叆锛屸渓og鈥濃4鈥濃溍封濃渓og鈥濃2鈥濄3銆佹渶鍚庝竴姝ワ紝鎸変笅绛変簬鍙峰氨鑳藉嚭鐜扮粨鏋滀簡銆
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...
绛旓細鎸囨暟鐨杩愮畻娉曞垯锛1銆乕a^m]脳[a^n]=a^(m锛媙) 銆愬悓搴曟暟骞傜浉涔,搴曟暟涓嶅彉,鎸囨暟鐩稿姞銆2銆乕a^m]梅[a^n]=a^(m锛峮) 銆愬悓搴曟暟骞傜浉闄,搴曟暟涓嶅彉,鎸囨暟鐩稿噺銆3銆乕a^m]^n=a^(mn) 銆愬箓鐨勪箻鏂,搴曟暟涓嶅彉,鎸囨暟鐩镐箻銆4銆乕ab]^m=(a^m)脳(a^m) 銆愮Н鐨勪箻鏂,绛変簬鍚勪釜鍥犲紡鍒嗗埆涔樻柟,鍐嶆妸鎵...
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瀽锛屼粙缁嶅浣璁＄畻瀵规暟銆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...
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...
绛旓細渚嬪lg10=1,lg100=lg102=2,lg4000=lg锛103脳4锛=3+lg4,鍙鍙瀵规煇涓鑼冨洿鐨勬暟缂栧埗鍑瀵规暟琛紝渚垮彲鍒╃敤鏉璁＄畻鍏朵粬鍗佽繘鍒舵暟鐨勫鏁扮殑杩戜技鍊笺傚湪鏁板鐞嗚涓婁竴鑸兘鐢ㄤ互鏃犵悊鏁癳=2.7182818鈥︹︿负搴曠殑瀵规暟锛屽苟灏嗚鍙 loge銆傜畝鍐欎负ln锛岀О涓鸿嚜鐒跺鏁帮紝鍥犱负鑷劧瀵规暟鍑芥暟鐨勫鏁拌〃杈惧紡鐗瑰埆绠娲侊紝鎵浠ユ樉鍑轰簡...
绛旓細log鍑芥暟杩愮畻鍏紡鎹㈠簳鍏紡浠嬬粛濡備笅锛歭oga(N)=x锛屽垯 a^x=N锛屼袱杈瑰彇浠涓哄簳鐨勫鏁帮紝logb(a^x)=logb(N)锛寈logb(a)=logb(N)锛寈=logb(N)/logb(a)锛屾墍浠oga(N)=logb(N)/logb(a)銆傛崲搴曞叕寮忔槸楂樹腑鏁板甯哥敤瀵规暟杩愮畻鍏紡锛屽彲灏嗗寮傚簳瀵规暟寮忚浆鍖栦负鍚屽簳瀵规暟寮忥紝缁撳悎鍏朵粬鐨勫鏁拌繍绠楀叕寮忎竴璧蜂娇鐢...
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杩愮畻鐨勯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...
