对数的换底公式是什么啊？ 对数的换底公式是什么？
\u5bf9\u6570\u7684\u6362\u5e95\u516c\u5f0f\u662f\u4ec0\u4e48\uff1f\u6362\u5e95\u516c\u5f0f\u662f\u4e00\u4e2a\u6bd4\u8f83\u91cd\u8981\u7684\u516c\u5f0f\uff0c\u5728\u5f88\u591a\u5bf9\u6570\u7684\u8ba1\u7b97\u4e2d\u90fd\u8981\u4f7f\u7528\uff0c\u4e5f\u662f\u9ad8\u4e2d\u6570\u5b66\u7684\u91cd\u70b9\u3002
log\uff08a\uff09\uff08b\uff09\u8868\u793a\u4ee5a\u4e3a\u5e95\u7684b\u7684\u5bf9\u6570\u3002
\u6240\u8c13\u7684\u6362\u5e95\u516c\u5f0f\u5c31\u662f
log
a
b=log(n)(b)/log(n)\uff08a\uff09
\u6362\u5e95\u516c\u5f0f
\u6362\u5e95\u516c\u5f0f\u662f
\u9ad8\u4e2d\u6570\u5b66\u5e38\u7528\u5bf9\u6570\u8fd0\u7b97\u516c\u5f0f\uff0c\u53ef\u5c06\u591a\u5f02\u5e95
\u5bf9\u6570\u5f0f\u8f6c\u5316\u4e3a\u540c\u5e95\u5bf9\u6570\u5f0f\uff0c\u7ed3\u5408\u5176\u4ed6\u7684\u5bf9\u6570\u8fd0\u7b97\u516c\u5f0f\u4e00\u8d77\u4f7f\u7528\u3002\u8ba1\u7b97\u4e2d\u5e38\u5e38\u4f1a\u51cf\u5c11\u8ba1\u7b97\u7684\u96be\u5ea6\uff0c\u66f4\u8fc5\u901f\u7684\u89e3\u51b3\u9ad8\u4e2d\u8303\u56f4\u7684\u5bf9\u6570\u8fd0\u7b97\u3002
\u6362\u5e95\u516c\u5f0f\u662f\u4e00\u4e2a\u6bd4\u8f83\u91cd\u8981\u7684\u516c\u5f0f\uff0c\u5728\u5f88\u591a\u5bf9\u6570\u7684\u8ba1\u7b97\u4e2d\u90fd\u8981\u4f7f\u7528\uff0c\u4e5f\u662f\u9ad8\u4e2d\u6570\u5b66\u7684\u91cd\u70b9\u3002
log\uff08a\uff09\uff08b\uff09\u8868\u793a\u4ee5a\u4e3a\u5e95\u7684b\u7684\u5bf9\u6570\u3002
\u6240\u8c13\u7684\u6362\u5e95\u516c\u5f0f\u5c31\u662f
log
a
b=log(n)(b)/log(n)\uff08a\uff09
\u6362\u5e95\u516c\u5f0f
\u6362\u5e95\u516c\u5f0f\u662f
\u9ad8\u4e2d\u6570\u5b66\u5e38\u7528\u5bf9\u6570\u8fd0\u7b97\u516c\u5f0f\uff0c\u53ef\u5c06\u591a\u5f02\u5e95
\u5bf9\u6570\u5f0f\u8f6c\u5316\u4e3a\u540c\u5e95\u5bf9\u6570\u5f0f\uff0c\u7ed3\u5408\u5176\u4ed6\u7684\u5bf9\u6570\u8fd0\u7b97\u516c\u5f0f\u4e00\u8d77\u4f7f\u7528\u3002\u8ba1\u7b97\u4e2d\u5e38\u5e38\u4f1a\u51cf\u5c11\u8ba1\u7b97\u7684\u96be\u5ea6\uff0c\u66f4\u8fc5\u901f\u7684\u89e3\u51b3\u9ad8\u4e2d\u8303\u56f4\u7684\u5bf9\u6570\u8fd0\u7b97\u3002
loga(b)表示以a为底的b的对数。
换底公式就是
loga(b)=logc(b)/logc(a)(a,c均大于零且不等于1）
绛旓細瀵规暟鎹㈠簳鍏紡锛log(a)(N)=log(b)(N)梅log(b)(a)銆傝繍绠楁硶鍒 loga(MN)=logaM+logaN loga(M/N)=logaM-logaN logaNn=nlogaN (n,M,N鈭圧)濡傛灉a=em锛屽垯m涓烘暟a鐨勮嚜鐒跺鏁帮紝鍗砽na=m锛宔=2.718281828?涓鸿嚜鐒跺鏁扮殑搴曪紝鍏朵负鏃犻檺涓嶅惊鐜皬鏁般傚畾涔夛細鑻n=b锛坅>0锛宎鈮1锛夊垯n=logab銆傛崲搴曞叕...
绛旓細瀵规暟鎹㈠簳鍏紡锛歭og(a)(N)=log(b)(N)梅log(b)(a)銆傝繍绠楁硶鍒欙細1銆乴oga(MN)=logaM+logaN銆2銆乴oga(M/N)=logaM-logaN銆3銆乴ogaNn=nlogaN銆4銆(n,M,N鈭圧)銆傚鏋渁=em锛屽垯m涓烘暟a鐨勮嚜鐒跺鏁帮紝鍗砽na=m锛宔=2.718281828鈥︿负鑷劧瀵规暟鐨勫簳锛屽叾涓烘棤闄愪笉寰幆灏忔暟銆傚畾涔夛細鑻n=b锛坅>0锛宎鈮1...
绛旓細瀵规暟鎹㈠簳鍏紡涓猴細log = ln / ln銆傚叾涓紝log琛ㄧず浠ヤ换鎰忓簳鏁癮鐨勫鏁帮紝ln琛ㄧず浠ヨ嚜鐒舵暟搴曟暟e鐨勫鏁帮紝b鏄崲搴曞叕寮忕殑搴曟暟銆傝繖鏉″叕寮忓厑璁告垜浠牴鎹凡鐭ョ殑瀵规暟鍑芥暟鏉ヨ绠楀叾浠栧簳鏁扮殑瀵规暟銆傚鏁版崲搴曞叕寮忕殑璇︾粏瑙ｉ噴濡備笅锛氬鏁版崲搴曞叕寮忔槸涓绉嶆暟瀛﹀叕寮忥紝瀹冭鏄庝簡鍦ㄤ换浣曞鏁拌绠椾腑锛岄兘鍙互浣跨敤涓涓笉鍚岀殑搴曟暟鏉ヨ〃杩般
绛旓細logab鎹㈠簳鍏紡鏄痩ogab=logca/logcb銆傚鏁版崲搴曞叕寮忕畝绉版崲搴曞叕寮忥紝鏄鏁扮殑涓绉嶆亽绛夊彉褰紝鎸囨洿鎹㈠簳鏁版椂鍚屼竴鐪熸暟鐨勪袱涓鏁伴棿鐨勫叧绯诲紡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ㄣ傝绠椾腑甯稿父浼氬噺灏戣绠楃殑闅惧害锛屾洿杩呴熺殑瑙ｅ喅楂樹腑鑼冨洿鐨勫鏁拌繍绠...
绛旓細鏍规嵁鏌ヨ鐖遍棶鏁欒偛寰楃煡锛屽鏁扮殑鎹㈠簳鍏紡锛log(a)(x)=log(b)(x)/log(b)(a)=lg(x)/lg(a)=ln(x)/ln(a)銆傚鏁拌繍绠楁硶鍒欙紝鏄竴绉嶇壒娈婄殑杩愮畻鏂规硶銆傛寚绉佸晢銆佸箓銆佹柟鏍圭殑瀵规暟鐨勮繍绠楁硶鍒欍傚湪鏁板涓紝瀵规暟鏄姹傚箓鐨勯嗚繍绠楋紝姝ｅ闄ゆ硶鏄箻娉曠殑鍊掓暟锛屽弽涔嬩害鐒躲傝繖鎰忓懗鐫涓涓暟瀛楃殑瀵规暟锛屾槸蹇呴』浜х敓...
绛旓細log鎹㈠簳鍏紡鏄細logₐb=logₘb/logₘa銆俵og鎹㈠簳鍏紡鏄鏁拌繍绠椾腑鐨勪竴涓噸瑕佸伐鍏凤紝瀹冨厑璁告垜浠湪涓嶅悓搴曟暟涔嬮棿杩涜杞崲锛屼粠鑰岀畝鍖栦簡璁＄畻杩囩▼銆俵ogₐb=logₘb/logₘa涓璦銆乥鍜宮閮芥槸姝ｅ疄鏁帮紝涓攁鍜宮閮戒笉绛変簬1銆傝繖涓叕寮忚〃绀轰互a涓哄簳b鐨勫鏁扮瓑浜庝互m涓哄簳b鐨勫鏁...
绛旓細a鈮1锛宐>0锛宐鈮1涓擭>0鏃讹紝logbN=logₐN/logₐb锛岀О涓哄鏁版崲搴曞叕寮忥紝寮忎腑1/logab绉颁负浠涓哄簳鐨勫鏁版崲鎴愪互b涓哄簳鐨勫鏁扮殑杞崲妯°傚鏁版崲搴曞叕寮(formula of change of base of logarithms)绠绉版崲搴曞叕寮忥紝鏄鏁扮殑涓绉嶆亽绛夊彉褰紝鎸囨洿鎹㈠簳鏁版椂鍚屼竴鐪熸暟鐨勪袱涓鏁伴棿鐨勫叧绯诲紡銆
绛旓細瀵规暟鍑芥暟鎹㈠簳鍏紡锛loga锛坆锛=logc锛坆锛/logc锛坅锛夛紙c锛0锛宑鈮1锛夋帹瀵艰繃绋嬶細浠oga锛坆锛=x 鍗砤^x=b,涓よ竟鍙栦互c锛坈锛0锛宑鈮1锛変负搴曠殑瀵规暟锛宭ogc锛坅^x锛=logc锛坆锛夊嵆x logc锛坅锛=logc锛坆锛夋晠鐢盿鈮1锛屽嵆 logc锛坅锛夆墵0 鍗硏=logc锛坆锛/ logc锛坅锛夋墍浠ワ紝loga锛坆锛=logc锛坆锛/logc...
绛旓細log鐨勫鏁版崲搴曞叕寮忔槸锛歭ogₐb = logcb / logca锛屽叾涓璦,b,c涓烘瀹炴暟涓攁鈮1锛宑鈮1锛宑鈮0銆
绛旓細鎹㈠簳鍏紡鏄楂樹腑鏁板甯哥敤瀵规暟杩愮畻鍏紡锛屽彲灏嗗寮搴曞鏁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ㄣ傝绠椾腑甯稿父浼氬噺灏戣绠楃殑闅惧害锛屾洿杩呴熺殑瑙ｅ喅楂樹腑鑼冨洿鐨勫鏁拌繍绠椼傛崲搴曞叕寮忓簲鐢細鍦ㄥ伐绋嬫妧鏈腑锛屾崲搴曞叕寮忎篃鏄粡甯哥敤鍒扮殑鍏紡锛屼緥濡傦紝鍦ㄧ紪绋嬭瑷涓紝鏈変簺缂栫▼璇█锛堜緥濡侰璇█锛夋病鏈変互a涓哄簳b涓虹湡鏁...
