{"id":956,"date":"2023-11-18T23:08:08","date_gmt":"2023-11-18T15:08:08","guid":{"rendered":"http:\/\/ggapa.net:81\/?p=956"},"modified":"2023-11-20T22:54:00","modified_gmt":"2023-11-20T14:54:00","slug":"%e5%af%b9%e6%95%b0","status":"publish","type":"post","link":"http:\/\/ggapa.net:81\/2023\/11\/18\/%e5%af%b9%e6%95%b0\/","title":{"rendered":"\u5bf9\u6570"},"content":{"rendered":"<h1>1 \u4ec0\u4e48\u662f\u5bf9\u6570<\/h1>\n<p>\u5982\u679c $a^x = N(a&gt;0, a \\neq 1, N&gt;0)$ \u5219 $x$ \u53eb\u505a\u4ee5 $a$ \u4e3a\u5e95 $N$ \u7684\u5bf9\u6570\uff0c\u8bb0\u505a $x=\\log_aN$ , \u5176\u4e2d $N$ \u88ab\u79f0\u4e3a\u771f\u6570\u3002<\/p>\n<p>\u82e5\u5bf9\u6570\u7684\u5e95\u6570\u4e3a $10$ \u5219\u8bb0\u4f5c $\\lg N$\uff0c\u82e5\u5bf9\u6570\u7684\u5e95\u6570\u4e3a\u81ea\u7136\u5e95\u6570 $e$\uff0c\u5219\u8bb0\u4f5c $\\ln N$ \u3002<\/p>\n<h1>2 \u5bf9\u6570\u7684\u8fd0\u7b97\u6cd5\u5219<\/h1>\n<h2>2.1 \u5bf9\u6570\u7684\u57fa\u672c\u8fd0\u7b97<\/h2>\n<ol>\n<li>$log_a M + log_aN = log_aMN$ <\/li>\n<li>$log_aM - log_aN = log_a\\frac{M}{N}$<\/li>\n<li>$\\log_ab^x=x\\log_ab$ <\/li>\n<\/ol>\n<h2>2.2 \u6362\u5e95\u516c\u5f0f<\/h2>\n<p>\u57fa\u672c\u516c\u5f0f\uff1a$\\frac{\\log_cb}{\\log_ca} = \\log_ab ,c&gt;0, c\\neq1,a\\neq1$<\/p>\n<blockquote>\n<p>\u8bc1\u660e\uff1a\u5df2\u77e5 $\\log_ab = x \\Leftrightarrow a^x = b$  \u5219 $\\log_c a^x = \\log_c b$ <\/p>\n<p>\u7531\u5bf9\u6570\u7684\u57fa\u672c\u8fd0\u7b97 $3$ \u53ef\u77e5\uff1a$x\\log_ca = log_cb$ <\/p>\n<p>\u6545 $x = \\frac{\\log_cb}{\\log_ca} = \\log_ab$<\/p>\n<\/blockquote>\n<p>\u6269\u5c551\uff1a$\\frac{1}{\\log_ba} = \\log_ab,b\\neq 1$ <\/p>\n<blockquote>\n<p>\u8bc1\u660e\uff1a\u5df2\u77e5 $\\frac{\\log_cb}{\\log_ca} = \\log_ab$<\/p>\n<p>\u5f53 $c=b$ \u65f6, $\\frac{1}{\\log_ba} = \\log_ab$<\/p>\n<\/blockquote>\n<p>\u6269\u5c552\uff1a$\\log_{a^x}b^y = \\frac{y}{x}\\cdot\\log_ab$<\/p>\n<blockquote>\n<p>\u8bc1\u660e\uff1a$\\log_{a^x}b^y=y\\log_{a^x}b = y \\cdot \\frac{1}{\\log_ba^x} = y \\cdot \\frac{1}{x\\log_b^a} = \\frac{y}{x}\\cdot\\log_ab$<\/p>\n<\/blockquote>\n<hr \/>\n<p>$\\mathrm{{\\Large Example1} } $<\/p>\n<p><strong>\u95ee\u9898\u63cf\u8ff0<\/strong><\/p>\n<p>\u6c42 $\\log_48$ \u7684\u503c\u3002<\/p>\n<p><strong>\u5206\u6790\u4e0e\u89e3\u7b54<\/strong><\/p>\n<ul>\n<li>\u65b9\u6cd5\u4e00\uff1a\u5229\u7528\u6362\u5e95\u516c\u5f0f\u53ef\u77e5 $\\log_48 = \\frac{\\log_28}{\\log_24} = \\frac{3}{2}$<\/li>\n<li>\u65b9\u6cd5\u4e8c\uff1a\u5229\u7528\u6362\u5e95\u516c\u5f0f\u6269\u5c55\u4e8c\u53ef\u76f4\u63a5\u6c42\u51fa\u7b54\u6848 $\\frac{3}{2}$<\/li>\n<\/ul>\n<p>$\\mathrm{{\\Large Example2} } $<\/p>\n<p><strong>\u9898\u76ee\u63cf\u8ff0<\/strong><\/p>\n<p>\u8bc1\u660e\uff1a $\\log_ab \\cdot\\log_bc\\cdot \\log_ca = 1$<br \/>\n<strong>\u5206\u6790\u4e0e\u89e3\u7b54<\/strong><\/p>\n<p>\u6b64\u9898\u6700\u5927\u7684\u96be\u70b9\u4e3a\u6bcf\u4e2a\u5bf9\u6570\u7684\u5e95\u6570\u4e0d\u540c\uff0c\u4e3a\u4e86\u4f7f\u4ed6\u4eec\u7684\u5e95\u6570\u76f8\u540c\u53ef\u4f7f\u7528\u57fa\u672c\u6362\u5e95\u516c\u5f0f\u3002<\/p>\n<p>\u539f\u5f0f\u53ef\u5316\u4e3a $\\log_ab \\cdot \\frac{\\log_ac}{\\log_ab} \\cdot  a\\frac{\\log_aa}{\\log_ac} = \\log_aa = 1$ <\/p>\n<h2>3.2 \u5bf9\u6570\u51fd\u6570\u7684\u56fe\u50cf<\/h2>\n<p>\u5bf9\u4e8e\u51fd\u6570 $y=\\log_ax$ ,$a^y = x$\uff1a<\/p>\n<ul>\n<li>\u5f53 $a &gt; 1 $ \u65f6\uff0c$y$ \u968f\u7740 $x$ \u7684\u589e\u5927\u800c\u589e\u5927\uff1b\u5f53 $y=0$ \u65f6\uff0c $x = 1$\u3002<\/li>\n<li>\u5f53 $a &lt;  1$ \u65f6\uff0c$y$ \u968f\u7740 $x$ \u7684\u589e\u5927\u800c\u51cf\u5c0f\uff1b\u5f53 $y=0$ \u65f6\uff0c $x = 1$\u3002<\/li>\n<\/ul>\n<p>\u82e5\u4e24\u4e2a\u5bf9\u6570\u51fd\u6570\u7684\u5e95\u6570\u4e92\u4e3a\u5012\u6570\uff0c\u5219\u4e24\u4e2a\u51fd\u6570\u5173\u4e8e $x$ \u8f74\uff0c\u5bf9\u79f0\u3002<\/p>\n<p>\u51fd\u6570\u7684\u5e95\u6570\u8d8a\u5927\uff0c\u51fd\u6570\u7684\u589e\u957f\u901f\u5ea6\u8d8a\u6162\u3002<\/p>\n<p>$\\mathrm{{\\Large Example1} } $<\/p>\n<p><strong>\u95ee\u9898\u63cf\u8ff0<\/strong><\/p>\n<p>\u5df2\u77e5 $a=2^{1.1},b=\\log_23,c=3^{\\log_3\\frac{3}{2}}$ ,\u6c42 $a,b,c$ \u7684\u5927\u5c0f\u5173\u7cfb\u3002<\/p>\n<p><strong>\u5206\u6790\u4e0e\u89e3\u7b54<\/strong><\/p>\n<p>\u6b64\u9898\u7684\u5173\u952e\u5728\u4e8e\u4f30\u7b97\u3002<\/p>\n<p>$a = 2^{1.1} \\approx 2^1 \\approx  2$<\/p>\n<p>$b = \\log_23 \\Leftrightarrow 2^b = 3 \\Rightarrow b &lt; 2 &lt; a$<\/p>\n<p>$c =3^{ \\log_3\\frac{3}{2} } = \\frac{3}{2} = \\log_22^\\frac{3}{2} = \\log_2\\sqrt{8} &lt; \\log_23$ <\/p>\n<p>\u6545 $a&gt;b&gt;c$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1 \u4ec0\u4e48\u662f\u5bf9\u6570 \u5982\u679c $a^x = N(a&gt;0, a \\neq 1, N&gt;0)$ \u5219 $x$ \u53eb\u505a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17],"tags":[26],"_links":{"self":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/956"}],"collection":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/comments?post=956"}],"version-history":[{"count":10,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/956\/revisions"}],"predecessor-version":[{"id":972,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/956\/revisions\/972"}],"wp:attachment":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/media?parent=956"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/categories?post=956"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/tags?post=956"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}