急求指数函数和对数函数的运算公式 指数函数的运算公式有哪些？ 怎么求指数函数和对数函数的定义域...
\u6307\u6570\u51fd\u6570\u548c\u5bf9\u6570\u51fd\u6570\u7684\u8fd0\u7b97\u516c\u5f0f1\u5bf9\u6570\u7684\u6982\u5ff5
\u5982\u679ca(a>0\uff0c\u4e14a\u22601)\u7684b\u6b21\u5e42\u7b49\u4e8eN\uff0c\u5373ab=N\uff0c\u90a3\u4e48\u6570b\u53eb\u505a\u4ee5a\u4e3a\u5e95N\u7684\u5bf9\u6570\uff0c\u8bb0\u4f5c\uff1alogaN=b,\u5176\u4e2da\u53eb\u505a\u5bf9\u6570\u7684\u5e95\u6570\uff0cN\u53eb\u505a\u771f\u6570.
\u7531\u5b9a\u4e49\u77e5\uff1a
\u2460\u8d1f\u6570\u548c\u96f6\u6ca1\u6709\u5bf9\u6570;
\u2461a>0\u4e14a\u22601,N>0;
\u2462loga1=0,logaa=1,alogaN=N,logaab=b.
\u7279\u522b\u5730\uff0c\u4ee510\u4e3a\u5e95\u7684\u5bf9\u6570\u53eb\u5e38\u7528\u5bf9\u6570\uff0c\u8bb0\u4f5clog10N,\u7b80\u8bb0\u4e3algN\uff1b\u4ee5\u65e0\u7406\u6570e(e=2.718 28\u2026)\u4e3a\u5e95\u7684\u5bf9\u6570\u53eb\u505a\u81ea\u7136\u5bf9\u6570\uff0c\u8bb0\u4f5clogeN\uff0c\u7b80\u8bb0\u4e3alnN.
2\u5bf9\u6570\u5f0f\u4e0e\u6307\u6570\u5f0f\u7684\u4e92\u5316
\u5f0f\u5b50\u540d\u79f0abN\u6307\u6570\u5f0fab=N(\u5e95\u6570)(\u6307\u6570)(\u5e42\u503c)\u5bf9\u6570\u5f0flogaN=b(\u5e95\u6570)(\u5bf9\u6570)(\u771f\u6570)
3\u5bf9\u6570\u7684\u8fd0\u7b97\u6027\u8d28
\u5982\u679ca>0,a\u22601,M>0,N>0,\u90a3\u4e48
(1)loga(MN)=logaM+logaN.
(2)logaMN=logaM-logaN.
(3)logaMn=nlogaM (n\u2208R).
\u95ee\uff1a\u2460\u516c\u5f0f\u4e2d\u4e3a\u4ec0\u4e48\u8981\u52a0\u6761\u4ef6a>0,a\u22601\uff0cM>0,N>0?
\u2461logaan=? (n\u2208R)
\u2462\u5bf9\u6570\u5f0f\u4e0e\u6307\u6570\u5f0f\u7684\u6bd4\u8f83.(\u5b66\u751f\u586b\u8868)
\u5f0f\u5b50ab=NlogaN=b\u540d\u79f0a\u2014\u5e42\u7684\u5e95\u6570
b\u2014
N\u2014a\u2014\u5bf9\u6570\u7684\u5e95\u6570
b\u2014
N\u2014\u8fd0
\u7b97
\u6027
\u8d28am\u00b7an=am+n
am\u00f7an=
(am)n=
(a>0\u4e14a\u22601,n\u2208R)logaMN=logaM+logaN
logaMN=
logaMn=(n\u2208R)
(a>0,a\u22601,M>0,N>0)
\u96be\u70b9\u7591\u70b9\u7a81\u7834
\u5bf9\u6570\u5b9a\u4e49\u4e2d\uff0c\u4e3a\u4ec0\u4e48\u8981\u89c4\u5b9aa\uff1e0,\uff0c\u4e14a\u22601?
\u7406\u7531\u5982\u4e0b\uff1a
\u2460\u82e5a\uff1c0\uff0c\u5219N\u7684\u67d0\u4e9b\u503c\u4e0d\u5b58\u5728\uff0c\u4f8b\u5982log-28
\u2461\u82e5a=0\uff0c\u5219N\u22600\u65f6b\u4e0d\u5b58\u5728\uff1bN=0\u65f6b\u4e0d\u60df\u4e00\uff0c\u53ef\u4ee5\u4e3a\u4efb\u4f55\u6b63\u6570
\u2462\u82e5a=1\u65f6\uff0c\u5219N\u22601\u65f6b\u4e0d\u5b58\u5728\uff1bN=1\u65f6b\u4e5f\u4e0d\u60df\u4e00\uff0c\u53ef\u4ee5\u4e3a\u4efb\u4f55\u6b63\u6570
\u4e3a\u4e86\u907f\u514d\u4e0a\u8ff0\u5404\u79cd\u60c5\u51b5\uff0c\u6240\u4ee5\u89c4\u5b9a\u5bf9\u6570\u5f0f\u7684\u5e95\u662f\u4e00\u4e2a\u4e0d\u7b49\u4e8e1\u7684\u6b63\u6570
\u89e3\u9898\u65b9\u6cd5\u6280\u5de7
1
(1)\u5c06\u4e0b\u5217\u6307\u6570\u5f0f\u5199\u6210\u5bf9\u6570\u5f0f\uff1a
\u246054=625\uff1b\u24612-6=164\uff1b\u24623x=27\uff1b\u246313m=573.
\uff082\uff09\u5c06\u4e0b\u5217\u5bf9\u6570\u5f0f\u5199\u6210\u6307\u6570\u5f0f\uff1a
\u2460log1216=-4\uff1b\u2461log2128=7\uff1b
\u2462log327=x\uff1b\u2463lg0.01=-2\uff1b
\u2464ln10=2.303\uff1b\u2465lg\u03c0=k.
\u89e3\u6790\u7531\u5bf9\u6570\u5b9a\u4e49\uff1aab=NlogaN=b.
\u89e3\u7b54(1)\u2460log5625=4.\u2461log2164=-6.
\u2462log327=x.\u2463log135.73=m.
\u89e3\u9898\u65b9\u6cd5
\u6307\u6570\u5f0f\u4e0e\u5bf9\u6570\u5f0f\u7684\u4e92\u5316\uff0c\u5fc5\u987b\u5e76\u4e14\u53ea\u9700\u7d27\u7d27\u6293\u4f4f\u5bf9\u6570\u7684\u5b9a\u4e49\uff1aab=NlogaN=b.(2)\u246012-4=16.\u246127=128.\u24623x=27.
\u246310-2=0.01.\u2464e2.303=10.\u246510k=\u03c0.
2
\u6839\u636e\u4e0b\u5217\u6761\u4ef6\u5206\u522b\u6c42x\u7684\u503c\uff1a
(1)log8x=-23\uff1b(2)log2(log5x)=0\uff1b
(3)logx27=31+log32\uff1b(4)logx(2+3)=-1.
\u89e3\u6790(1)\u5bf9\u6570\u5f0f\u5316\u6307\u6570\u5f0f\uff0c\u5f97\uff1ax=8-23=?
(2)log5x=20=1. x=?
(3)31+log32=3\u00d73log32=?27=x?
(4)2+3=x-1=1x. x=?
\u89e3\u7b54(1)x=8-23=(23)-23=2-2=14.
(2)log5x=20=1\uff0cx=51=5.
(3)logx27=3\u00d73log32=3\u00d72=6\uff0c
\u2234x6=27=33=(3)6,\u6545x=3.
(4)2+3=x-1=1x,\u2234x=12+3=2-3.
\u89e3\u9898\u6280\u5de7
\u2460\u8f6c\u5316\u7684\u601d\u60f3\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u6570\u5b66\u601d\u60f3\uff0c\u5bf9\u6570\u5f0f\u4e0e\u6307\u6570\u5f0f\u6709\u7740\u5bc6\u5207\u7684\u5173\u7cfb\uff0c\u5728\u89e3\u51b3\u6709\u5173\u95ee\u9898\u65f6\uff0c\u7ecf\u5e38\u8fdb\u884c\u7740\u4e24\u79cd\u5f62\u5f0f\u7684\u76f8\u4e92\u8f6c\u5316.
\u2461\u719f\u7ec3\u5e94\u7528\u516c\u5f0f\uff1aloga1=0,logaa=1,alogaM=M,logaan=n.3
\u5df2\u77e5logax=4,logay=5\uff0c\u6c42A=\u3014x\u00b73x-1y2\u301512\u7684\u503c.
\u89e3\u6790\u601d\u8def\u4e00\uff0c\u5df2\u77e5\u5bf9\u6570\u5f0f\u7684\u503c\uff0c\u8981\u6c42\u6307\u6570\u5f0f\u7684\u503c\uff0c\u53ef\u5c06\u5bf9\u6570\u5f0f\u8f6c\u5316\u4e3a\u6307\u6570\u5f0f\uff0c\u518d\u5229\u7528\u6307\u6570\u5f0f\u7684\u8fd0\u7b97\u6c42\u503c\uff1b
\u601d\u8def\u4e8c\uff0c\u5bf9\u6307\u6570\u5f0f\u7684\u4e24\u8fb9\u53d6\u540c\u5e95\u7684\u5bf9\u6570\uff0c\u518d\u5229\u7528\u5bf9\u6570\u5f0f\u7684\u8fd0\u7b97\u6c42\u503c
\u89e3\u7b54\u89e3\u6cd5\u4e00\u2235logax=4,logay=5,
\u2234x=a4,y=a5,
\u2234A=x512y-13=(a4)512(a5)-13=a53\u00b7a-53=a0=1.
\u89e3\u6cd5\u4e8c\u5bf9\u6240\u6c42\u6307\u6570\u5f0f\u4e24\u8fb9\u53d6\u4ee5a\u4e3a\u5e95\u7684\u5bf9\u6570\u5f97
logaA=loga(x512y-13)
=512logax-13logay=512\u00d74-13\u00d75=0,
\u2234A=1.
\u89e3\u9898\u6280\u5de7
\u6709\u65f6\u5bf9\u6570\u8fd0\u7b97\u6bd4\u6307\u6570\u8fd0\u7b97\u6765\u5f97\u65b9\u4fbf\uff0c\u56e0\u6b64\u4ee5\u6307\u6570\u5f62\u5f0f\u51fa\u73b0\u7684\u5f0f\u5b50\uff0c\u53ef\u5229\u7528\u53d6\u5bf9\u6570\u7684\u65b9\u6cd5\uff0c\u628a\u6307\u6570\u8fd0\u7b97\u8f6c\u5316\u4e3a\u5bf9\u6570\u8fd0\u7b97.4
\u8bbex,y\u5747\u4e3a\u6b63\u6570\uff0c\u4e14x\u00b7y1+lgx=1(x\u2260110),\u6c42lg(xy)\u7684\u53d6\u503c\u8303\u56f4.
\u89e3\u6790\u4e00\u4e2a\u7b49\u5f0f\u4e2d\u542b\u4e24\u4e2a\u53d8\u91cfx\u3001y\uff0c\u5bf9\u6bcf\u4e00\u4e2a\u786e\u5b9a\u7684\u6b63\u6570x\u7531\u7b49\u5f0f\u90fd\u6709\u60df\u4e00\u7684\u6b63\u6570y\u4e0e\u4e4b\u5bf9\u5e94\uff0c\u6545y\u662fx\u7684\u51fd\u6570\uff0c\u4ece\u800clg(xy)\u4e5f\u662fx\u7684\u51fd\u6570.\u56e0\u6b64\u6c42lg(xy)\u7684\u53d6\u503c\u8303\u56f4\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u6c42\u51fd\u6570\u503c\u57df\u7684\u95ee\u9898\uff0c\u600e\u6837\u624d\u80fd\u5efa\u7acb\u8fd9\u79cd\u51fd\u6570\u5173\u7cfb\u5462?\u80fd\u5426\u5bf9\u5df2\u77e5\u7684\u7b49\u5f0f\u4e24\u8fb9\u4e5f\u53d6\u5bf9\u6570?
\u89e3\u7b54\u2235x>0,y>0,x\u00b7y1+lgx=1,
\u4e24\u8fb9\u53d6\u5bf9\u6570\u5f97\uff1algx+(1+lgx)lgy=0.
\u5373lgy=-lgx1+lgx(x\u2260110,lgx\u2260-1).
\u4ee4lgx=t, \u5219lgy=-t1+t(t\u2260-1).
\u2234lg(xy)=lgx+lgy=t-t1+t=t21+t.
\u89e3\u9898\u89c4\u5f8b
\u5bf9\u4e00\u4e2a\u7b49\u5f0f\u4e24\u8fb9\u53d6\u5bf9\u6570\u662f\u89e3\u51b3\u542b\u6709\u6307\u6570\u5f0f\u548c\u5bf9\u6570\u5f0f\u95ee\u9898\u7684\u5e38\u7528\u7684\u6709\u6548\u65b9\u6cd5\uff1b\u800c\u53d8\u91cf\u66ff\u6362\u53ef\u628a\u8f83\u590d\u6742\u95ee\u9898\u8f6c\u5316\u4e3a\u8f83\u7b80\u5355\u7684\u95ee\u9898.\u8bbeS=t21+t,\u5f97\u5173\u4e8et\u7684\u65b9\u7a0bt2-St-S=0\u6709\u5b9e\u6570\u89e3.
\u2234\u0394=S2+4S\u22650\uff0c\u89e3\u5f97S\u2264-4\u6216S\u22650,
\u6545lg(xy)\u7684\u53d6\u503c\u8303\u56f4\u662f(-\u221e,-4\u3015\u222a\u30140,+\u221e).
5
\u6c42\u503c\uff1a
(1)lg25+lg2\u00b7lg50+(lg2)2\uff1b
(2)2log32-log3329+log38-52log53\uff1b
(3)\u8bbelga+lgb=2lg(a-2b)\uff0c\u6c42log2a-log2b\u7684\u503c;
(4)\u6c427lg20\u00b712lg0.7\u7684\u503c.
\u89e3\u6790(1)25=52,50=5\u00d710.\u90fd\u5316\u6210lg2\u4e0elg5\u7684\u5173\u7cfb\u5f0f.
(2)\u8f6c\u5316\u4e3alog32\u7684\u5173\u7cfb\u5f0f.
(3)\u6240\u6c42log2a-log2b=log2ab\u7531\u5df2\u77e5\u7b49\u5f0f\u7ed9\u51fa\u4e86a,b\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u80fd\u5426\u4ece\u4e2d\u6c42\u51faab\u7684\u503c\u5462?
(4)7lg20\u00b712lg0.7\u662f\u4e24\u4e2a\u6307\u6570\u5e42\u7684\u4e58\u79ef\uff0c\u4e14\u6307\u6570\u542b\u5e38\u7528\u5bf9\u6570\uff0c
\u8bbex=7lg20\u00b712lg0.7\u80fd\u5426\u5148\u6c42\u51falgx\uff0c\u518d\u6c42x?
\u89e3\u7b54(1)\u539f\u5f0f=lg52+lg2\u00b7lg(10\u00d75)+(lg2)2
=2lg5+lg2\u00b7(1+lg5)+(lg2)2
=lg5\u00b7(2+lg2)+lg2+(lg2)2
=lg102\u00b7(2+lg2)+lg2+(lg2)2
=(1-lg2)(2+lg2)+lg2+(lg2)2
=2-lg2-(lg2)2+lg2+(lg2)2=2.
(2)\u539f\u5f0f=2log32-(log325-log332)+log323-5log59
=2log32-5log32+2+3log32-9
=-7.
(3)\u7531\u5df2\u77e5lgab=lg(a-2b)2 (a-2b>0),
\u2234ab=(a-2b)2, \u5373a2-5ab+4b2=0.
\u2234ab=1\u6216ab=4\uff0c\u8fd9\u91cca>0,b>0.
\u82e5ab=1\uff0c\u5219a-2b<0, \u2234ab=1\uff08 \u820d\u53bb\uff09.
\u2234ab=4,
\u2234log2a-log2b=log2ab=log24=2.
(4)\u8bbex=7lg20\u00b712lg0.7,\u5219
lgx=lg20\u00d7lg7+lg0.7\u00d7lg12 =(1+lg2)\u00b7lg7+(lg7-1)\u00b7(-lg2)
=lg7+lg2=14,
\u2234x=14, \u6545\u539f\u5f0f=14.
\u89e3\u9898\u89c4\u5f8b
\u2460\u5bf9\u6570\u7684\u8fd0\u7b97\u6cd5\u5219\u662f\u8fdb\u884c\u540c\u5e95\u7684\u5bf9\u6570\u8fd0\u7b97\u7684\u4f9d\u636e\uff0c\u5bf9\u6570\u7684\u8fd0\u7b97\u6cd5\u5219\u662f\u7b49\u5f0f\u4e24\u8fb9\u90fd\u6709\u610f\u4e49\u7684\u6052\u7b49\u5f0f\uff0c\u8fd0\u7528\u6cd5\u5219\u8fdb\u884c\u5bf9\u6570\u53d8\u5f62\u65f6\u8981\u6ce8\u610f\u5bf9\u6570\u7684\u771f\u6570\u7684\u8303\u56f4\u662f\u5426\u6539\u53d8\uff0c\u4e3a\u9632\u6b62\u589e\u6839\u6240\u4ee5\u9700\u8981\u68c0\u9a8c\uff0c\u5982(3).
\u2461\u5bf9\u4e00\u4e2a\u5f0f\u5b50\u5148\u6c42\u5b83\u7684\u5e38\u7528\u5bf9\u6570\u503c\uff0c\u518d\u6c42\u539f\u5f0f\u7684\u503c\u662f\u4ee3\u6570\u8fd0\u7b97\u4e2d\u5e38\u7528\u7684\u65b9\u6cd5\uff0c\u5982(4).6
\u8bc1\u660e(1)logaN=logcNlogca(a>0,a\u22601,c>0,c\u22601,N>0)\uff1b
(2)logab\u00b7logbc=logac\uff1b
(3)logab=1logba(b>0,b\u22601)\uff1b
(4)loganbm=mnlogab.
\u89e3\u6790(1)\u8bbelogaN=b\u5f97ab=N,\u4e24\u8fb9\u53d6\u4ee5c\u4e3a\u5e95\u7684\u5bf9\u6570\u6c42\u51fab\u5c31\u53ef\u80fd\u5f97\u8bc1.
(2)\u4e2dlogbc\u80fd\u5426\u4e5f\u6362\u6210\u4ee5a\u4e3a\u5e95\u7684\u5bf9\u6570.
(3)\u5e94\u7528(1)\u5c06logab\u6362\u6210\u4ee5b\u4e3a\u5e95\u7684\u5bf9\u6570.
(4)\u5e94\u7528(1)\u5c06loganbm\u6362\u6210\u4ee5a\u4e3a\u5e95\u7684\u5bf9\u6570.
\u89e3\u7b54(1)\u8bbelogaN=b\uff0c\u5219ab=N,\u4e24\u8fb9\u53d6\u4ee5c\u4e3a\u5e95\u7684\u5bf9\u6570\u5f97\uff1ab\u00b7logca=logcN,
\u2234b=logcNlogca.\u2234logaN=logcNlogca.
(2)\u7531(1)logbc=logaclogab.
\u6240\u4ee5 logab\u00b7logbc=logab\u00b7logaclogab=logac.
(3)\u7531(1)logab=logbblogba=1logba.
\u89e3\u9898\u89c4\u5f8b
(1)\u4e2dlogaN=logcNlogca\u53eb\u505a\u5bf9\u6570\u6362\u5e95\u516c\u5f0f\uff0c(2)(3)(4)\u662f(1)\u7684\u63a8\u8bba\uff0c\u5b83\u4eec\u5728\u5bf9\u6570\u8fd0\u7b97\u548c\u542b\u5bf9\u6570\u7684\u7b49\u5f0f\u8bc1\u660e\u4e2d\u7ecf\u5e38\u5e94\u7528.\u5bf9\u4e8e\u5bf9\u6570\u7684\u6362\u5e95\u516c\u5f0f\uff0c\u65e2\u8981\u5584\u4e8e\u6b63\u7528\uff0c\u4e5f\u8981\u5584\u4e8e\u9006\u7528.(4)\u7531(1)loganbm=logabmlogaan=mlogabnlogaa=mnlogab.
7
\u5df2\u77e5log67=a,3b=4,\u6c42log127.
\u89e3\u6790\u4f9d\u9898\u610fa,b\u662f\u5e38\u6570\uff0c\u6c42log127\u5c31\u662f\u8981\u7528a,b\u8868\u793alog127\uff0c\u53c83b=4\u5373log34=b\uff0c\u80fd\u5426\u5c06log127\u8f6c\u5316\u4e3a\u4ee56\u4e3a\u5e95\u7684\u5bf9\u6570\uff0c\u8fdb\u800c\u8f6c\u5316\u4e3a\u4ee53\u4e3a\u5e95\u5462?
\u89e3\u7b54\u5df2\u77e5log67=a,log34=b,
\u2234log127=log67log612=a1+log62.
\u53c8log62=log32log36=log321+log32,
\u7531log34=b,\u5f972log32=b.
\u2234log32=b2,\u2234log62=b21+b2=b2+b.
\u2234log127=a1+b2+b=a(2+b)2+2b.
\u89e3\u9898\u6280\u5de7
\u5229\u7528\u5df2\u77e5\u6761\u4ef6\u6c42\u5bf9\u6570\u7684\u503c\uff0c\u4e00\u822c\u8fd0\u7528\u6362\u5e95\u516c\u5f0f\u548c\u5bf9\u6570\u8fd0\u7b97\u6cd5\u5219\uff0c\u628a\u5bf9\u6570\u7528\u5df2\u77e5\u6761\u4ef6\u8868\u793a\u51fa\u6765\uff0c\u8fd9\u662f\u5e38\u7528\u7684\u65b9\u6cd5\u6280\u5de78
\u5df2\u77e5x,y,z\u2208R+\uff0c\u4e143x=4y=6z.
(1)\u6c42\u6ee1\u8db32x=py\u7684p\u503c\uff1b
(2)\u6c42\u4e0ep\u6700\u63a5\u8fd1\u7684\u6574\u6570\u503c\uff1b
(3)\u6c42\u8bc1\uff1a12y=1z-1x.
\u89e3\u6790\u5df2\u77e5\u6761\u4ef6\u4e2d\u7ed9\u51fa\u4e86\u6307\u6570\u5e42\u7684\u8fde\u7b49\u5f0f\uff0c\u80fd\u5426\u5f15\u8fdb\u4e2d\u95f4\u91cfm\uff0c\u518d\u7528m\u5206\u522b\u8868\u793ax,y,z?\u53c8\u60f3\uff0c\u5bf9\u4e8e\u6307\u6570\u5f0f\u80fd\u5426\u7528\u5bf9\u6570\u7684\u65b9\u6cd5\u53bb\u89e3\u7b54?
\u89e3\u7b54\uff081\uff09\u89e3\u6cd5\u4e003x=4ylog33x=log34yx=ylog342x=2ylog34=ylog316,
\u2234p=log316.
\u89e3\u6cd5\u4e8c\u8bbe3x=4y=m,\u53d6\u5bf9\u6570\u5f97\uff1a
x\u00b7lg3=lgm\uff0cylg4=lgm,
\u2234x=lgmlg3,y=lgmlg4,2x=2lgmlg3,py=plgmlg4.
\u75312y=py, \u5f97 2lgmlg3=plgmlg4,
\u2234p=2lg4lg3=lg42lg3=log316.
(2)\u22352=log39<log316<log327=3,
\u22342<p<3.
\u53c83-p=log327-log316=log32716,
p-2=log316-log39=log3169,
\u800c2716<169,
\u2234log327163-p.
\u2234\u4e0ep\u6700\u63a5\u8fd1\u7684\u6574\u6570\u662f3.
\u89e3\u9898\u601d\u60f3
\u2460\u63d0\u5021\u4e00\u9898\u591a\u89e3.\u4e0d\u540c\u7684\u601d\u8def\uff0c\u4e0d\u540c\u7684\u65b9\u6cd5\uff0c\u5e94\u7528\u4e86\u4e0d\u540c\u7684\u77e5\u8bc6\u6216\u8005\u662f\u76f8\u540c\u77e5\u8bc6\u7684\u7075\u6d3b\u8fd0\u7528\uff0c\u65e2\u53d1\u6563\u4e86\u601d\u7ef4\uff0c\u53c8\u63d0\u9ad8\u4e86\u5206\u6790\u95ee\u9898\u548c\u89e3\u51b3\u95ee\u9898\u7684\u80fd\u529b\uff0c\u4f55\u4e50\u800c\u4e0d\u4e3a\u5462?
\u2461(2)\u4e2d\u6d89\u53ca\u6bd4\u8f83\u4e24\u4e2a\u5bf9\u6570\u7684\u5927\u5c0f.\u8fd9\u662f\u540c\u5e95\u7684\u4e24\u4e2a\u5bf9\u6570\u6bd4\u5927\u5c0f.\u56e0\u4e3a\u5e953>1\uff0c\u6240\u4ee5\u771f\u6570\u5927\u7684\u5bf9\u6570\u5c31\u5927\uff0c\u95ee\u9898\u8f6c\u5316\u4e3a\u6bd4\u8f83\u4e24\u4e2a\u771f\u6570\u7684\u5927\u5c0f\uff0c\u8fd9\u91cc\u8d85\u524d\u5e94\u7528\u4e86\u5bf9\u6570\u51fd\u6570\u7684\u5355\u8c03\u6027\uff0c\u4ee5\u9f13\u52b1\u5b66\u751f\u8d85\u524d\u5b66\u4e60\uff0c\u81ea\u89c9\u5b66\u4e60\u7684\u5b66\u4e60\u79ef\u6781\u6027.(3)\u89e3\u6cd5\u4e00\u4ee43x=4y=6z=m,\u7531\u4e8ex\uff0cy\uff0cz\u2208R+\uff0c
\u2234k>1\uff0c\u5219 x=lgmlg3,y=lgmlg4,z=lgmlg6,
\u6240\u4ee51z-1x=lg6lgm-lg3lgm=lg6-lg3lgm=lg2lgm\uff0c12y=12\u00b7lg4lgm=lg2lgm\uff0c
\u654512y=1z-1x.
\u89e3\u6cd5\u4e8c3x=4y=6z=m\uff0c
\u5219\u67093=m1x\u2460,4=m1y\u2461\uff0c6=m1z\u2462\uff0c
\u2462\u00f7\u2460\uff0c\u5f97m1z-1x=63=2=m12y.
\u22341z-1x=12y.
9
\u5df2\u77e5\u6b63\u6570a,b\u6ee1\u8db3a2+b2=7ab.\u6c42\u8bc1\uff1alogma+b3=12(logma+logmb)(m>0\u4e14m\u22601).
\u89e3\u6790\u5df2\u77e5a>0,b>0,a2+b2=7ab.\u6c42\u8bc1\u5f0f\u4e2d\u771f\u6570\u90fd\u53ea\u542ba,b\u7684\u4e00\u6b21\u5f0f\uff0c\u60f3\uff1a\u80fd\u5426\u5c06\u771f\u6570\u4e2d\u7684\u4e00\u6b21\u5f0f\u4e5f\u8f6c\u5316\u4e3a\u4e8c\u6b21\uff0c\u8fdb\u800c\u5e94\u7528a2+b2=7ab?
\u89e3\u7b54logma+b3=logm(a+b3)212=
\u89e3\u9898\u6280\u5de7
\u2460\u5c06a+b3\u5411\u4e8c\u6b21\u8f6c\u5316\u4ee5\u5229\u4e8e\u5e94\u7528a2+b2=7ab\u662f\u6280\u5de7\u4e4b\u4e00.
\u2461\u5e94\u7528a2+b2=7ab\u5c06\u771f\u6570\u7684\u548c\u5f0f\u8f6c\u5316\u4e3aab\u7684\u4e58\u79ef\u5f0f\uff0c\u4ee5\u4fbf\u4e8e\u5e94\u7528\u5bf9\u6570\u8fd0\u7b97\u6027\u8d28\u662f\u6280\u5de7\u4e4b\u4e8c.12logma+b32=12logma2+b2+2ab9.
\u2235a2+b2=7ab,
\u2234logma+b3=12logm7ab+2ab9=12logmab=12(logma+logmb),
\u5373logma+b3=12(logma+logmb).
\u601d\u7ef4\u62d3\u5c55\u53d1\u6563
1
\u6570\u5b66\u5174\u8da3\u5c0f\u7ec4\u4e13\u95e8\u7814\u7a76\u4e86\u79d1\u5b66\u8bb0\u6570\u6cd5\u4e0e\u5e38\u7528\u5bf9\u6570\u95f4\u7684\u5173\u7cfb.\u8bbe\u771f\u6570N=a\u00d710n.\u5176\u4e2dN>0,1\u2264a<10,n\u2208Z.\u8fd9\u5c31\u662f\u7528\u79d1\u5b66\u8bb0\u6570\u6cd5\u8868\u793a\u771f\u6570N.\u5176\u79d1\u5b66\u6027\u4f53\u73b0\u5728\u54ea\u91cc?\u6211\u4eec\u53ea\u8981\u7814\u7a76\u6570N\u7684\u5e38\u7528\u5bf9\u6570\uff0c\u5c31\u80fd\u63ed\u793a\u5176\u4e2d\u7684\u5965\u79d8.
\u89e3\u6790\u7531\u5df2\u77e5\uff0c\u5bf9N=a\u00d710n\u53d6\u5e38\u7528\u5bf9\u6570\u5f97\uff0clgN=n+lga.\u771f\u6570\u4e0e\u5bf9\u6570\u6709\u4f55\u8054\u7cfb?
\u89e3\u7b54lgN=lg(a\u00d710n)=n+lga.n\u2208Z,1\u2264a<10\uff0c
\u2234lga\u2208\u30140,1).
\u6211\u4eec\u628a\u6574\u6570n\u53eb\u505aN\u7684\u5e38\u7528\u5bf9\u6570\u7684\u9996\u6570\uff0c\u628alga\u53eb\u505aN\u7684\u5e38\u7528\u5bf9\u6570\u7684\u5c3e\u6570\uff0c\u5b83\u662f\u6b63\u7684\u7eaf\u5c0f\u6570\u62160.
\u5c0f\u7ed3\uff1a\u2460lgN\u7684\u9996\u6570\u5c31\u662fN\u4e2d10n\u7684\u6307\u6570\uff0c\u5c3e\u6570\u5c31\u662flga,0\u2264lga<1;
\u2461\u6709\u6548\u6570\u5b57\u76f8\u540c\u7684\u4e0d\u540c\u6b63\u6570\u5b83\u4eec\u7684\u5e38\u7528\u5bf9\u6570\u7684\u5c3e\u6570\u76f8\u540c\uff0c\u53ea\u662f\u9996\u6570\u4e0d\u540c\uff1b
\u2462\u5f53N\u22651\u65f6\uff0clgN\u7684\u9996\u6570n\u6bd4\u5b83\u7684\u6574\u6570\u4f4d\u6570\u5c111\uff0c\u5f53N\u2208(0\uff0c1)\u65f6\uff0clgN\u7684\u9996\u6570n\u662f\u8d1f\u6574\u6570\uff0c|n|-1\u4e0eN\u7684\u5c0f\u6570\u70b9\u540e\u7b2c\u4e00\u4e2a\u4e0d\u662f0\u7684\u6709\u6548\u6570\u5b57\u524d\u7684\u96f6\u7684\u4e2a\u6570\u76f8\u540c.
\u5e08\u751f\u4e92\u52a8
\u4ec0\u4e48\u53eb\u505a\u79d1\u5b66\u8bb0\u6570\u6cd5\uff1f
N>0,lgN\u7684\u9996\u6570\u548c\u5c3e\u6570\u4e0ea\u00d710n\u6709\u4ec0\u4e48\u8054\u7cfb\uff1f
\u6709\u6548\u6570\u5b57\u76f8\u540c\u7684\u4e0d\u540c\u6b63\u6570\u5176\u5e38\u7528\u5bf9\u6570\u7684\u4ec0\u4e48\u76f8\u540c\uff1f\u4ec0\u4e48\u4e0d\u540c\uff1f
2
\u82e5lgx\u7684\u9996\u6570\u6bd4lg1x\u7684\u9996\u6570\u59279\uff0clgx\u7684\u5c3e\u6570\u6bd4lg1x\u7684\u5c3e\u6570\u5c0f0380 4\uff0c\u4e14lg0.203 4=1.308 3,\u6c42lgx,x,lg1x\u7684\u503c.
\u89e3\u6790\u2460lg0.203 4=1308 3,\u5373lg0.203 4=1+0.308 3\uff0c1\u662f\u5bf9\u6570\u7684\u9996\u6570\uff0c0.308 3\u662f\u5bf9\u6570\u7684\u5c3e\u6570\uff0c\u662f\u6b63\u7684\u7eaf\u5c0f\u6570\uff1b\u2461\u82e5\u8bbelgx=n+lga\uff0c\u5219lg1x\u4e5f\u53ef\u8868\u51fa.
\u89e3\u7b54\u8bbelgx=n+lga,\u4f9d\u9898\u610flg1x=(n-9)+(lga+0.380 4).
\u53c8lg1x=-lgx=-(n+lga),
\u2234(n-9)+(lga+0380 4)=-n-lga\uff0c\u5176\u4e2dn-9\u662f\u9996\u6570\uff0clga+0380 4\u662f\u5c3e\u6570\uff0c-n-lga=-(n+1)+(1-lga),-(n+1)\u662f\u9996\u65701-lga\u662f\u5c3e\u6570\uff0c\u6240\u4ee5\uff1a
n-9=-(n+1)
lga+0.380 4=1-lgan=4,
lga=0.308 3.
\u2234lgx=4+0.308 3=4.308 3,
\u2235lg0.203 4=1.308 3,\u2234x=2.034\u00d7104.
\u2234lg1x=-(4+0.308 3)=5.691 7.
\u89e3\u9898\u89c4\u5f8b
\u628algx\u7684\u9996\u6570\u548c\u5c3e\u6570\uff0clg1x\u7684\u9996\u6570\u548c\u5c3e\u6570\u90fd\u770b\u6210\u672a\u77e5\u6570\uff0c\u6839\u636e\u9898\u76ee\u7684\u7b49\u91cf\u5173\u7cfb\u5217\u65b9\u7a0b.\u518d\u7531\u540c\u4e00\u5bf9\u6570\u7684\u9996\u6570\u7b49\u4e8e\u9996\u6570\uff0c\u5c3e\u6570\u7b49\u4e8e\u5c3e\u6570\uff0c\u6c42\u51fa\u672a\u77e5\u6570\u7684\u503c\uff0c\u662f\u89e3\u51b3\u8fd9\u7c7b\u95ee\u9898\u7684\u5e38\u7528\u65b9\u6cd5.3
\u8ba1\u7b97\uff1a
(1)log2-3(2+3)+log6(2+3+2-3);
(2)2lg(lga100)2+lg(lga).
\u89e3\u6790(1)\u4e2d.2+3\u4e0e2-3\u6709\u4f55\u5173\u7cfb?2+3+2-3\u53cc\u91cd\u6839\u53f7\uff0c\u5982\u4f55\u5316\u7b80?
(2)\u4e2d\u5206\u6bcd\u5df2\u65e0\u6cd5\u5316\u7b80\uff0c\u5206\u5b50\u80fd\u5316\u7b80\u5417?
\u89e3\u9898\u65b9\u6cd5
\u8ba4\u771f\u5ba1\u9898\u3001\u7406\u89e3\u9898\u610f\u3001\u6293\u4f4f\u7279\u70b9\u3001\u627e\u51fa\u660e\u786e\u7684\u89e3\u9898\u601d\u8def\u548c\u65b9\u6cd5\uff0c\u4e0d\u8981\u88ab\u8868\u9762\u7684\u7e41\u3001\u96be\u6240\u5413\u5012.\u89e3\u7b54(1)\u539f\u5f0f=log2-3(2-3\uff09-1+12log6(2+3+2-3)2
=-1+12log6(4+22+3\u00b72-3)
=-1+12log66
=-12.
(2)\u539f\u5f0f=2lg(100lga)2+lg(lga)=2\u3014lg100+lg(lga)\u30152+lg(lga)=2\u30142+lg(lga)\u30152+lg(lga)=2.
4
\u5df2\u77e5log2x=log3y=log5z<0,\u6bd4\u8f83x,3y,5z\u7684\u5927\u5c0f.
\u89e3\u6790\u5df2\u77e5\u662f\u5bf9\u6570\u7b49\u5f0f\uff0c\u8981\u6bd4\u8f83\u5927\u5c0f\u7684\u662f\u6839\u5f0f\uff0c\u6839\u5f0f\u80fd\u8f6c\u5316\u6210\u6307\u6570\u5e42\uff0c\u6240\u4ee5\uff0c\u5bf9\u6570\u7b49\u5f0f\u5e94\u8bbe\u6cd5\u8f6c\u5316\u4e3a\u6307\u6570\u5f0f.
\u89e3\u7b54\u8bbelog2x=log3y=log5z=m<0.\u5219
x=2m,y=3m,z=5m.
x=(2)m,3y=(33)m,5z=(55)m.
\u4e0b\u9762\u53ea\u9700\u6bd4\u8f832\u4e0e33,55\u7684\u5927\u5c0f\uff1a
(2)6=23=8,(33)6=32=9\uff0c\u6240\u4ee52<33.
\u53c8(2)10=25=32,(55)10=52=25,
\u22342>55.
\u223455<2<33. \u53c8m<0,
\u56fe2-7-1\u8003\u67e5\u6307\u6570\u51fd\u6570y=(2)x,y=(33)x,y=(55)x\u5728\u7b2c\u4e8c\u8c61\u9650\u7684\u56fe\u50cf\uff0c\u5982\u56fe2-7-1
\u89e3\u9898\u89c4\u5f8b
\u2460\u8f6c\u5316\u7684\u601d\u60f3\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u6570\u5b66\u601d\u60f3\uff0c\u5bf9\u6570\u4e0e\u6307\u6570\u6709\u7740\u5bc6\u5207\u7684\u5173\u7cfb\uff0c\u5728\u89e3\u51b3\u6709\u5173\u95ee\u9898\u65f6\u8981\u5145\u5206\u6ce8\u610f\u8fd9\u79cd\u5173\u7cfb\u53ca\u5bf9\u6570\u5f0f\u4e0e\u6307\u6570\u5f0f\u7684\u76f8\u4e92\u8f6c\u5316.
\u2461\u6bd4\u8f83\u6307\u6570\u76f8\u540c\uff0c\u5e95\u4e0d\u540c\u7684\u6307\u6570\u5e42(\u5e95\u5927\u4e8e0)\u7684\u5927\u5c0f\uff0c\u8981\u5e94\u7528\u591a\u4e2a\u6307\u6570\u51fd\u6570\u5728\u540c\u4e00\u5750\u6807\u7cfb\u4e2d\u7b2c\u4e00\u8c61\u9650(\u6307\u6570\u5927\u4e8e0)\u6216\u7b2c\u4e8c\u8c61\u9650(\u6307\u6570\u5c0f\u4e8e0)\u7684\u6027\u8d28\u8fdb\u884c\u6bd4\u8f83
\u2460\u662fy=(55)x,\u2461\u662fy=(2)x,\u2462\u662fy=(33)x.\u6307\u6570m<0\u65f6\uff0c\u56fe\u50cf\u5728\u7b2c\u4e8c\u8c61\u9650\u4ece\u4e0b\u5230\u4e0a\uff0c\u5e95\u4ece\u5927\u5230\u5c0f.\u6240\u4ee5(33)m<(2)m<(55)m,\u65453y<x<5z.
\u6f5c\u80fd\u6311\u6218\u6d4b\u8bd5
1(1)\u5c06\u4e0b\u5217\u6307\u6570\u5f0f\u5316\u4e3a\u5bf9\u6570\u5f0f:
\u246073=343;\u246114-2=16;\u2462e-5=m.
(2)\u5c06\u4e0b\u5217\u5bf9\u6570\u5f0f\u5316\u4e3a\u6307\u6570\u5f0f\uff1a
\u2460log128=-3;\u2461lg10000=4;\u2462ln3.5=p.
2\u8ba1\u7b97\uff1a
(1)24+log23;(2)2723-log32;(3)2513log527+2log52.
3(1)\u5df2\u77e5lg2=0.301 0\uff0clg3=0.477 1\uff0c\u6c42lg45;
(2)\u82e5lg3.127=a\uff0c\u6c42lg0.031 27.
4\u5df2\u77e5a\u22600,\u5219\u4e0b\u5217\u5404\u5f0f\u4e2d\u4e0elog2a2\u603b\u76f8\u7b49\u7684\u662f()
A\u82e5logx+1(x+1)=1 ,\u5219x\u7684\u53d6\u503c\u8303\u56f4\u662f()
A\u5df2\u77e5ab=M(a>0,b>0,M\u22601),\u4e14logMb=x\uff0c\u5219logMa\u7684\u503c\u4e3a()
A\u82e5log63=0.673 1\uff0clog6x=-0.326 9, \u5219x\u4e3a()
A\u82e5log5\u3014log3(log2x)\u3015=0\uff0c\u5219x=.
98log87\u00b7log76\u00b7log65=.
10\u5982\u679c\u65b9\u7a0blg2x+(lg2+lg3)lgx+lg2\u00b7lg3=0\u7684\u4e24\u6839\u4e3ax1\u3001x2,\u90a3\u4e48x1\u00b7x2\u7684\u503c\u4e3a.
11\u751f\u6001\u5b66\u6307\u51fa\uff1a\u751f\u7269\u7cfb\u7edf\u4e2d\uff0c\u6bcf\u8f93\u5165\u4e00\u4e2a\u8425\u517b\u7ea7\u7684\u80fd\u91cf\uff0c\u5927\u7ea6\u53ea\u670910%\u7684\u80fd\u91cf\u6d41\u5230\u4e0b\u4e00\u4e2a\u8425\u517b\u7ea7.H1\u2192H2\u2192H3\u2192H4\u2192H5\u2192H6\u8fd9\u6761\u751f\u7269\u94fe\u4e2d(Hn\u8868\u793a\u7b2cn\u4e2a\u8425\u517b\u7ea7\uff0cn=1\uff0c2\uff0c3\uff0c4\uff0c5\uff0c6).\u5df2\u77e5\u5bf9H1\u8f93\u5165\u4e86106\u5343\u7126\u7684\u80fd\u91cf\uff0c\u95ee\u7b2c\u51e0\u4e2a\u8425\u517b\u7ea7\u80fd\u83b7\u5f97100\u5343\u7126\u7684\u80fd\u91cf?
12\u5df2\u77e5x\uff0cy\uff0cz\u2208R+\u4e143x=4y=6z\uff0c\u6bd4\u8f833x\uff0c4y\uff0c6z\u7684\u5927\u5c0f.
13\u5df2\u77e5a,b\u5747\u4e3a\u4e0d\u7b49\u4e8e1\u7684\u6b63\u6570\uff0c\u4e14axby=aybx=1\uff0c\u6c42\u8bc1x2=y2.
14\u5df2\u77e52a\u00b75b=2c\u00b75d=10\uff0c\u8bc1\u660e(a-1)(d-1)=(b-1)(c-1).
15\u8bbe\u96c6\u5408M=\uff5bx|lg\u3014ax2-2(a+1)x-1\u3015>0\uff5d\uff0c\u82e5M\u2260\uff0cM\uff5bx|x<0\uff5d\uff0c\u6c42\u5b9e\u6570a\u7684\u53d6\u503c\u8303\u56f4.
16\u5728\u5f20\u6c5f\u9ad8\u79d1\u6280\u56ed\u533a\u7684\u4e0a\u6d77\u8d85\u7ea7\u8ba1\u7b97\u4e2d\u5fc3\u5185\uff0c\u88ab\u79f0\u4e3a\u201c\u795e\u5a01\u2160\u201d\u7684\u8ba1\u7b97\u673a\u8fd0\u7b97\u901f\u5ea6\u4e3a\u6bcf\u79d2\u949f384 000 000 000\u6b21.\u7528\u79d1\u5b66\u8bb0\u6570\u6cd5\u8868\u793a\u8fd9\u4e2a\u6570\u4e3aN=,\u82e5\u5df2\u77e5lg3.840=0.584 3,\u5219lgN=.
17\u67d0\u5de5\u5382\u5f15\u8fdb\u65b0\u7684\u751f\u4ea7\u8bbe\u5907\uff0c\u9884\u8ba1\u4ea7\u54c1\u7684\u751f\u4ea7\u6210\u672c\u6bd4\u4e0a\u4e00\u5e74\u964d\u4f4e10%\uff0c\u8bd5\u95ee\u7ecf\u8fc7\u51e0\u5e74\uff0c\u751f\u4ea7\u6210\u672c\u964d\u4f4e\u4e3a\u539f\u6765\u768440%?(lg2=0.3, lg3=0.48)
18\u67d0\u5382\u4e3a\u9002\u5e94\u6539\u9769\u5f00\u653e\uff0c\u5b8c\u5584\u7ba1\u7406\u673a\u5236\uff0c\u6ee1\u8db3\u5e02\u573a\u9700\u6c42\uff0c\u67d0\u79cd\u4ea7\u54c1\u6bcf\u5b63\u5ea6\u5e73\u5747\u6bd4\u4e0a\u4e00\u5b63\u5ea6\u589e\u957f10.4%\uff0c\u90a3\u4e48\u7ecf\u8fc7y\u5b63\u5ea6\u589e\u957f\u5230\u539f\u6765\u7684x\u500d\uff0c\u5219\u51fd\u6570y=f(x)\u7684\u89e3\u6790\u5f0ff(x)=.
\u540d\u5e08\u52a9\u4f60\u6210\u957f
1.(1)\u2460log7343=3.\u2461log1416=-2.\u2462lnm=-5.
(2)\u246012-3=8.\u2461104=10 000.\u2462ep=3.5.
2.(1)48\u70b9\u62e8\uff1a\u5148\u5e94\u7528\u79ef\u7684\u4e58\u65b9\uff0c\u518d\u7528\u5bf9\u6570\u6052\u7b49\u5f0f.
(2)98\u70b9\u62e8\uff1a\u5e94\u7528\u5546\u7684\u4e58\u65b9\u548c\u5bf9\u6570\u6052\u7b49\u5f0f.
(3)144\u70b9\u62e8\uff1a\u5e94\u7528\u5bf9\u6570\u8fd0\u7b97\u6027\u8d28\u548c\u79ef\u7684\u4e58\u65b9.
3.(1)0.826 6\u70b9\u62e8\uff1alg45=12lg45=12lg902=12(lg32+lg10-lg2).
(2)lg0.031 27=lg(3.127\u00d710-2)=-2+lg3.127=-2+a
4.C\u70b9\u62e8\uff1aa\u22600,a\u53ef\u80fd\u662f\u8d1f\u6570\uff0c\u5e94\u7528\u5bf9\u6570\u8fd0\u7b97\u6027\u8d28\u8981\u6ce8\u610f\u5bf9\u6570\u90fd\u6709\u610f\u4e49.
5.B\u70b9\u62e8\uff1a\u5e95x+1>0\u4e14x+1\u22601;\u771f\u6570x+1>0.
6.A\u70b9\u62e8\uff1a\u5bf9ab=M\u53d6\u4ee5M\u4e3a\u5e95\u7684\u5bf9\u6570.
7.C\u70b9\u62e8\uff1a\u6ce8\u610f0.673 1+0.326 9=1,log61x=0.326 9\uff0c
\u6240\u4ee5log63+log61x=log63x=1.\u22343x=6, x=12.
8.x=8\u70b9\u62e8\uff1a\u7531\u5916\u5411\u5185.log3(log2x)=1, log2x=3, x=23.
9.5\u70b9\u62e8\uff1alog87\u00b7log76\u00b7log65=log85, 8log85=5.
10.16\u70b9\u62e8\uff1a\u5173\u4e8elgx\u7684\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\u7684\u4e24\u6839\u662flgx1,lgx2.
\u7531lgx1=-lg2,lgx2=-lg3\uff0c\u5f97x1=12,x2=13.
11.\u8bbe\u7b2cn\u4e2a\u8425\u517b\u7ea7\u80fd\u83b7\u5f97100\u5343\u7126\u7684\u80fd\u91cf\uff0c
\u4f9d\u9898\u610f:106\u00b710100n-1=100,
\u5316\u7b80\u5f97:107-n=102,\u5229\u7528\u540c\u5e95\u5e42\u76f8\u7b49\uff0c\u5f977-n=2,
\u6216\u8005\u4e24\u8fb9\u53d6\u5e38\u7528\u5bf9\u6570\u4e5f\u5f977-n=2.
\u2234n=5,\u5373\u7b2c5\u4e2a\u8425\u517b\u7ea7\u80fd\u83b7\u80fd\u91cf100\u5343\u7126.
12\u8bbe3x=4y=6z=k,\u56e0\u4e3ax\uff0cy\uff0cz\u2208R+\uff0c
\u6240\u4ee5k>1.\u53d6\u4ee5k\u4e3a\u5e95\u7684\u5bf9\u6570\uff0c\u5f97\uff1a
x=1logk3,y=1logk4,z=1logk6.
\u22343x=3logk3=113logk3=1logk33,
\u540c\u7406\u5f97\uff1a4y=1logk44,6z=1logk66.
\u800c33=1281,44=1264,66=1236,
\u2234logk33>logk44>logk66.
\u53c8k>1,33>44>66>1,
\u2234logk33>logk44>logk66>0,\u22343x<4y<6z.
13.\u2235axby=aybx=1,\u2234lg(axby)=lg(aybx)=0,
\u5373xlga+ylgb=ylga+xlgb=0.(\u203b)
\u4e24\u5f0f\u76f8\u52a0,\u5f97x(lga+lgb)+y(lga+lgb)=0.
\u5373(lga+lgb)(x+y)=0.\u2234lga+lgb=0 \u6216x+y=0.
\u5f53lga+lgb=0\u65f6,\u4ee3\u5165xlga+ylgb=0,\u5f97:
(x-y)lga=0, a\u662f\u4e0d\u4e3a1\u7684\u6b63\u6570lga\u22600,\u2234x-y=0.
\u2234x+y=0\u6216x-y=0,\u2234x2=y2.
14.\u22352a5b=10,\u22342a-1=51-b.\u4e24\u8fb9\u53d6\u4ee52\u4e3a\u5e95\u7684\u5bf9\u6570,\u5f97\uff1aa-1=(1-b)log25.
\u2234log25=a-11-b(b\u22601). \u540c\u7406\u5f97log25=c-11-d(d\u22601).
\u5373b\u22601,d\u22601\u65f6,a-11-b=c-11-d.
\u2234(a-1)(1-d)=(c-1)(1-b),
\u2234(a-1)(d-1)=(b-1)(c-1).
\u5f53b=1,c=1\u65f6\u663e\u7136\u6210\u7acb.
15.\u8bbelg\u3014ax2-2(a+1)x-1\u3015=t (t>0),\u5219
ax2-2(a+1)x-1=10t(t>0).
\u223410t>1 ,ax2-2(a+1)x-1>1,\u2234ax2-2(a+1)x-2>0.
\u2460\u5f53a=0\u65f6,\u89e3\u96c6\uff5bx|x<-1\uff5d\uff5bx|x<0\uff5d;
\u5f53a\u22600\u65f6,M\u2260\u4e14M\uff5bx|x<0\uff5d.
\u2234\u65b9\u7a0bax2-2(a+1)x-2=0 \u5fc5\u6709\u4e24\u4e0d\u7b49\u5b9e\u6839\uff0c\u8bbe\u4e3ax1,x2\u4e14x1<x2\uff0c\u5219
\u2461\u5f53a>0\u65f6,M=\uff5bx|xx2\uff5d,\u663e\u7136\u4e0d\u662f\uff5bx|x<0\uff5d\u7684\u5b50\u96c6\uff1b
\u2462\u5f53a<0\u65f6\uff0cM=\uff5bx|x1<x<x2\uff5d\u53ea\u8981\uff1a
a<0\uff0c
\u0394=4(a+1)2+8a>0\uff0c
x1+x2=2(a+1)a<0\uff0c
x1\u00b7x2=-2a>0.
\u89e3\u5f973-2<a<0\uff0c\u7efc\u4e0a\u6240\u6c42\uff0ca\u7684\u53d6\u503c\u8303\u56f4\u662f\uff1a3-2<a\u22640.
16.N=3.840\u00d71011, lgN=11.584 3.
17.\u8bbe\u7ecf\u8fc7x\u5e74\uff0c\u6210\u672c\u964d\u4e3a\u539f\u6765\u768440%.\u5219
(1-10%)x=40%,\u4e24\u8fb9\u53d6\u5e38\u7528\u5bf9\u6570\uff0c\u5f97\uff1a
x\u00b7lg(1-10%)=lg40% \uff0c
\u5373x=lg0.4lg0.9=lg4-1lg9-1=2lg2-12lg3-1=10.
\u6240\u4ee5\u7ecf\u8fc710\u5e74\u6210\u672c\u964d\u4f4e\u4e3a\u539f\u6765\u768440%.
18.f(x)=log1.104x\u3014\u6216f(x)=lgxlg1.104\u3015.
\u70b9\u62e8\uff1a\u8bbe\u539f\u6765\u4e00\u4e2a\u5b63\u5ea6\u4ea7\u54c1\u4e3aa\uff0c\u5219a(1+10.4%)y=xa,\u2234y=log1.10
\u53e6\u5916\u53c2\u770b\u8fd9\u4e2a\u516c\u5f0f\u3002\u5bf9\u6570\u51fd\u6570\u8fd0\u7b97\u516c\u5f0f
http://wenku.baidu.com/view/dc8f161b227916888486d75c.html
e\u7684\u5b9a\u4e49\uff1ae=lim(x\u2192\u221e)(1+1/x)^x=2.718281828... \u3000\u3000\u8bbea>0,a!=1----(log a(x))' \u3000\u3000=lim(\u0394x\u2192\u221e)((log a(x+\u0394x)-log a(x))/\u0394x) \u3000\u3000=lim(\u0394x\u2192\u221e)(1/x*x/\u0394x*log a((x+\u0394x)/x)) \u3000\u3000=lim(\u0394x\u2192\u221e)(1/x*log a((1+\u0394x/x)^(x/\u0394x))) \u3000\u3000=1/x*lim(\u0394x\u2192\u221e)(log a((1+\u0394x/x)^(x/\u0394x))) \u3000\u3000=1/x*log a(lim(\u0394x\u21920)(1+\u0394x/x)^(x/\u0394x)) \u3000\u3000=1/x*log a(e)\u7279\u6b8a\u5730\uff0c \u3000\u3000\u5f53a=e\u65f6\uff0c \u3000\u3000(log a(x))'=(ln x)'=1/x\u3002 \u3000\u3000\u8bbey=a^x\u4e24\u8fb9\u53d6\u5bf9\u6570ln y=xln a\u4e24\u8fb9\u5bf9\u6c42x \u3000\u3000\u5bfcy'/y=ln ay'=yln a=a^xln a\u7279\u6b8a\u5730\uff0c \u3000\u3000\u5f53a=e\u65f6\uff0cy'=(a^x)'=(e^x)'=e^xln e=e^x\u3002
\u5b9a\u4e49\u57df\uff1a\u5b9e\u6570\u96c6
\u3000\u3000\u6307\u4ee3\u4e00\u5207\u5b9e\u6570\uff08-\u221e\uff0c+\u221e\uff09\uff0c\u5c31\u662fR\u3002
\u7f16\u8f91\u672c\u6bb5\u503c\u57df\uff1a\uff080\uff0c+\u221e\uff09
\u3000\u3000\u5bf9\u4e8e\u4e00\u5207\u6307\u6570\u51fd\u6570y=a^x\u6765\u8bb2\u3002\u4ed6\u7684a\u6ee1\u8db3a>0\u4e14a\u22601\uff0c\u5373\u8bf4\u660ey>0\u3002\u6240\u4ee5\u503c\u57df\u4e3a\uff080,+\u221e)
\u7f16\u8f91\u672c\u6bb5\u5206\u5f0f\u5316\u7b80\u7684\u65b9\u6cd5\u4e0e\u6280\u5de7
\u3000\u3000\uff081\uff09\u628a\u5206\u5b50\u3001\u5206\u6bcd\u5206\u89e3\u56e0\u5f0f\uff0c\u53ef\u7ea6\u5206\u7684\u5148\u7ea6\u5206 \u3000\u3000\uff082\uff09\u5229\u7528\u516c\u5f0f\u7684\u57fa\u672c\u6027\u8d28\uff0c\u5316\u7e41\u5206\u5f0f\u4e3a\u7b80\u5206\u5f0f\uff0c\u5316\u5f02\u5206\u6bcd\u4e3a\u540c\u5206\u6bcd \u3000\u3000\uff083\uff09\u628a\u5176\u4e2d\u9002\u5f53\u7684\u51e0\u4e2a\u5206\u5f0f\u5148\u5316\u7b80\uff0c\u91cd\u70b9\u7a81\u7834. \u6307\u6570\u51fd\u6570
\uff084\uff09\u53ef\u8003\u8651\u6574\u4f53\u601d\u60f3\uff0c\u7528\u6362\u5143\u6cd5\u4f7f\u5206\u5f0f\u7b80\u5316
绛旓細璁鎸囨暟鍑芥暟涓簓=a^x 鍒欒浆鎹㈡垚瀵规暟鍑芥暟鏄痽=loga锛坸锛夋寚鏁板嚱鏁板悎鍜屼粬鐩稿簲鐨勫鏁板嚱鏁板簲璇ユ槸浜掍负鍙嶅嚱鏁 锛1+n锛塣7=10 鍙眰寰梟=log7锛10锛-1 鏈夋椂瀵规暟杩愮畻姣鎸囨暟杩愮畻鏉ュ緱鏂逛究锛屽洜姝や互鎸囨暟褰㈠紡鍑虹幇鐨勫紡瀛愶紝鍙埄鐢ㄥ彇瀵规暟鐨勬柟娉曪紝鎶婃寚鏁拌繍绠楄浆鍖栦负瀵规暟杩愮畻銆
绛旓細浣垮緱g(x)=loga(f(x))=log2(8)銆傛牴鎹簰鎹㈠叕寮忓彲浠ュ緱鍒癴(x)=a^x=8锛岃В寰梐=2锛屾墍浠(x)=log2(8)瀵瑰簲鐨勬寚鏁板嚱鏁版槸f(x)=2^x銆傜患涓婃墍杩帮紝鎸囨暟鍑芥暟鍜屽鏁板嚱鏁涔嬮棿瀛樺湪浜掍负鍙鍑芥暟鐨浜掓崲鍏紡锛岃兘澶熶簰鐩歌浆鍖栥傝繖绉嶄簰鎹㈠叕寮忓湪鏁板鍜岀瀛璁＄畻涓叿鏈夐噸瑕佺殑搴旂敤浠峰硷紝鍙互绠鍖栬绠楄繃绋嬪拰瑙ｅ喅闂銆
绛旓細鎸囨暟鍑芥暟 鎸囨暟鍑芥暟鐨涓鑸舰寮忎负y=a^x(a>0涓斾笉=1) ,浠庝笂闈㈡垜浠浜庡箓鍑芥暟鐨勮璁哄氨鍙互鐭ラ亾,瑕佹兂浣垮緱x鑳藉鍙栨暣涓疄鏁伴泦鍚堜负瀹氫箟鍩,鍒欏彧鏈変娇寰 濡傚浘鎵绀轰负a鐨勪笉鍚屽ぇ灏忓奖鍝嶅嚱鏁板浘褰㈢殑鎯呭喌.鍦ㄥ嚱鏁皔=a^x涓彲浠ョ湅鍒帮細锛1锛 鎸囨暟鍑芥暟鐨勫畾涔夊煙涓烘墍鏈夊疄鏁扮殑闆嗗悎,杩欓噷鐨勫墠鎻愭槸a澶т簬0涓斾笉绛変簬1,瀵逛簬a涓...
绛旓細璁鎸囨暟鍑芥暟涓簓=a^x 鍒欒浆鎹㈡垚瀵规暟鍑芥暟鏄痽=loga锛坸锛夋寚鏁板嚱鏁板悎鍜屼粬鐩稿簲鐨勫鏁板嚱鏁板簲璇ユ槸浜掍负鍙嶅嚱鏁 锛1+n锛塣7=10 鍙眰寰梟=log7锛10锛-1
绛旓細(7)瀵规暟鎭掔瓑寮忥細a^log锛坅)N=N;log锛坅锛塧^b=b 锛8锛夌敱骞傜殑瀵规暟鐨勮繍绠鎬ц川鍙緱(鎺ㄥ鍏紡)1.log(a)M^(1/n)=(1/n)log(a)M ,log(a)M^(-1/n)=(-1/n)log(a)M 2.log(a)M^(m/n)=(m/n)log(a)M ,log(a)M^(-m/n)=(-m/n)log(a)M 3.log(a^n)M^n=log(a)M...
绛旓細鎸囨暟鍜屽鏁鐨勮浆鎹㈠叕寮忔槸锛歛^y=xy=log(a)(x)銆瀵规暟鍑芥暟鐨涓鑸舰寮忎负 y=logax锛屽畠瀹為檯涓婂氨鏄鎸囨暟鍑芥暟鐨勫弽鍑芥暟锛屽浘璞″叧浜庣洿绾縴=x瀵圭О鐨勪袱鍑芥暟浜掍负鍙嶅嚱鏁帮紝鍙〃绀轰负x=a^y銆傚洜姝ゆ寚鏁板嚱鏁伴噷瀵逛簬a瀛樺湪瑙勫畾锛歛>0涓攁鈮1锛屽浜庝笉鍚屽ぇ灏廰浼氬舰鎴愪笉鍚岀殑鍑芥暟鍥惧舰鍏充簬X杞村绉般佸綋a>1鏃讹紝a瓒婂ぇ锛屽浘鍍忚秺...
绛旓細鍥犳鎸囨暟鍑芥暟閲屽浜巃瀛樺湪瑙勫畾鈥斺攁>0涓攁鈮1锛屽浜庝笉鍚屽ぇ灏廰浼氬舰鎴愪笉鍚岀殑鍑芥暟鍥惧舰锛氬叧浜嶺杞村绉般佸綋a>1鏃讹紝a瓒婂ぇ锛屽浘鍍忚秺闈犺繎x杞淬佸綋0<a<1鏃讹紝a瓒婂皬锛屽浘鍍忚秺闈犺繎x杞淬瀵规暟锛氬湪鏁板涓紝瀵规暟鏄姹傚箓鐨勯杩愮畻锛屾濡傞櫎娉曟槸涔樻硶鐨勯嗚繍绠楋紝鍙嶄箣浜︾劧銆傝繖鎰忓懗鐫涓涓暟瀛楃殑瀵规暟鏄繀椤讳骇鐢熷彟涓涓浐瀹...
绛旓細鎸囨暟锛氬姞鍑忔病浠涔堝ソ璇寸殑,鍜屽椤瑰紡鏄竴鏍风殑.涔橀櫎娉曪細鍒嗗埆鏄鎸囨暟鐨鐩稿姞鍜岀浉鍑,渚嬪e^x * e^2x=e^(x+2x)=e^3x,闄ゆ硶鍒欎负鐩稿噺.瀵规暟锛氬叾瀹瀵规暟鍜屾寚鏁鏄嗙潃鏉ョ殑,鎸囨暟涔樻硶鏄寚鏁扮浉鍔,瀵规暟鍔犳硶鍒欏氨鏄浉涔,鍑忔硶鍒欎负鐩搁櫎.渚嬪ln x+ln 2x=ln锛坸*2x锛=ln锛2x^2).
绛旓細鎸囨暟鍜屽鏁鐨勮浆鎹㈠叕寮忔槸锛歛^y=xy=log(a)(x)銆瀵规暟鍑芥暟鐨涓鑸舰寮忎负 y=logax锛屽畠瀹為檯涓婂氨鏄鎸囨暟鍑芥暟鐨勫弽鍑芥暟锛屽浘璞″叧浜庣洿绾縴=x瀵圭О鐨勪袱鍑芥暟浜掍负鍙嶅嚱鏁帮紝鍙〃绀轰负x=a^y銆傚洜姝ゆ寚鏁板嚱鏁伴噷瀵逛簬a瀛樺湪瑙勫畾锛歛>0涓攁鈮1锛屽浜庝笉鍚屽ぇ灏廰浼氬舰鎴愪笉鍚岀殑鍑芥暟鍥惧舰鍏充簬X杞村绉般佸綋a>1鏃讹紝a瓒婂ぇ锛屽浘鍍忚秺...
绛旓細褰揳>0涓攁鈮1鏃讹紝M>0,N>0锛岄偅涔堬細 锛1锛塴og(a)(MN)=log(a)(M)+log(a)(N); 锛2锛塴og(a)(M/N)=log(a)(M)-log(a)(N); 锛3锛塴og(a)(M^n)=nlog(a)(M) 锛坣鈭圧锛 (4)log(a^n)(M)=1/nlog(a)(M)(n鈭圧) 锛5锛夋崲搴曞叕寮忥細log(A)...
