{"id":67387,"date":"2020-07-03T10:09:01","date_gmt":"2020-07-03T01:09:01","guid":{"rendered":"http:\/\/mechsys.tec.u-ryukyu.ac.jp\/~oshiro\/SiteList\/?p=67387"},"modified":"2020-07-05T19:55:08","modified_gmt":"2020-07-05T10:55:08","slug":"%e5%af%be%e6%95%b0%e9%96%a2%e9%80%a3%e3%81%ae%e5%85%ac%e5%bc%8f","status":"publish","type":"post","link":"http:\/\/mechsys.tec.u-ryukyu.ac.jp\/~oshiro\/SiteList\/2020\/07\/03\/67387\/","title":{"rendered":"\u5bfe\u6570\u95a2\u9023\u306e\u516c\u5f0f"},"content":{"rendered":"

\uff08\u95a2\u9023\u8a18\u4e8b\uff1a\u5e38\u7528\u5bfe\u6570\u306e\u8fd1\u4f3c\u5024\u306e\u7b97\u51fa<\/a>\uff09<\/p>\n

\n\u5bfe\u6570\u306f\u300c$x$\u3092\u4f55\u4e57\u3057\u305f\u3089$z$\u306b\u306a\u308b\u304b\u300d\u3092\u6c42\u3081\u308b\u3082\u306e\u3068\u6349\u3048\u308b\u3002
\n$z=x^y$\u3067\u306e$y$\u3092\u77e5\u308a\u305f\u3044\u5834\u5408\u306b$y=\\log_{x}z$\u3068\u66f8\u304f\u3002
\n$x$\u3092\u5e95\uff08base\uff09\u3001$z$\u3092\u771f\u6570(anti-logarithm\u3001\u9006\u5bfe\u6570)\u3068\u547c\u3076\u3002
\n\u5e95\u304c10\u306e\u5834\u5408\u3092\u5e38\u7528\u5bfe\u6570\uff08$\\log_{10}$\uff09\u3002\u5e95\u304c${\\rm e}$\u306e\u5834\u5408\u3092\u81ea\u7136\u5bfe\u6570\uff08natural logarithm\uff09\u3068\u547c\u3073\u3001$\\log_{\\rm e}$\u3092$\\ln$\u3068\u3082\u66f8\u304f\u3002<\/p>\n

\u516c\u5f0f<\/h2>\n
\n
    \n
  1. \n$z=x^y \\Leftrightarrow y=\\log_x z$,<\/p>\n

  2. \n$x^{\\log_x z}=x^y=z$\uff08\u51aa\u306e\u5e95\u3068\u6307\u6570\u306b\u3042\u308b\u5bfe\u6570\u306e\u5e95\u304c\u540c\u3058\u5834\u5408\u306f\u3001\u6307\u6570\u90e8\u306e\u5bfe\u6570\u306e\u771f\u6570\u306e\u307f\u306b\uff09,<\/p>\n
  3. \n$\\log_\\underline{x} \\underline{x}^y = \\log_{x}z =y$\uff08\u5bfe\u6570\u306e\u5e95\u3068\u771f\u6570\u306e\u51aa\u306e\u5e95\u304c\u540c\u3058\u5834\u5408\u306f\u3001\u771f\u6570\u306e\u6307\u6570\u306b\uff09,<\/p>\n

  4. \n$x^1=x \\Leftrightarrow \\log_x x = 1$, $x^0=1 \\Leftrightarrow \\log_x 1 = 0$,<\/p>\n

  5. \n$z^p=(x^y)^p=\\underbrace{x^y\\cdot x^y\\cdots x^y}_p=x^{\\overbrace{y+y+\\cdots+y}^p}=x^{py}$
    \n\u3000\u3000$\\Rightarrow\\log_x z^p=\\log_x {(x^y)}^p=\\log_x x^{py}=py=p\\log_x{z}$\uff08\u771f\u6570\u306e\u6307\u6570\u306f\u4fc2\u6570\u306b\u51fa\u305b\u308b\uff09<\/p>\n

  6. \n$z_1=x^a, z_2=x^b; z_1z_2=x^ax^b=x^{a+b}$
    \n\u3000\u3000$\\Leftrightarrow$
    \n\u3000\u3000$\\log z_1z_2=\\log x^ax^b=\\log x^{a+b}$$=(a+b)\\log x=a\\log x + b\\log x$$=\\log x^a + \\log x^b=\\log z_1 + \\log z_2$\u3002
    \n\u3000\u3000$\\log z_1\/z_2=\\log x^ax^{-b}=\\log x^{a-b}$$=(a-b)\\log x=a\\log x – b\\log x$$=\\log x^a – \\log x^b=\\log z_1 – \\log z_2$\u3002
    \n\u3000\u3000\uff08\u5bfe\u6570\u3092\u53d6\u308b\u3068\u3001\u4e57\u9664\u304c\u52a0\u6e1b\u7b97\u306b\uff09<\/p>\n

  7. \n$z=x^y$\u3067\u4e21\u8fba\u306e\u5bfe\u6570\u3092\u53d6\u308b\u3068\u3001$\\log_a z = \\log_a x^y = y\\log_a x \\Rightarrow y = \\displaystyle\\frac{\\log_a z}{\\log_a x}$\u3002
    \n\u4e0a\u5f0f\u3067\u3001\u5bfe\u6570\u306e\u5e95\u3092$x$\u3068\u3059\u308b\u3068$y = \\displaystyle\\frac{\\log_x z}{\\log_x x}=\\displaystyle\\frac{\\log_x z}{1}=\\log_x z$ $\\Rightarrow \\log_x z=\\displaystyle\\frac{\\log_a z}{\\log_a x}$ (\u5bfe\u6570\u306e\u5e95\u306e\u5909\u63db)
    \n\u9006\u306b\u3001\u5bfe\u6570\u306e\u5e95\u3092$z$\u3068\u3059\u308b\u3068\u3001$y = \\displaystyle\\frac{\\log_z z}{\\log_z x}=\\displaystyle\\frac{1}{\\log_z x}$ $\\Rightarrow \\log_x z=\\displaystyle\\frac{1}{\\log_z x}$\uff08\u5e95\u3068\u771f\u6570\u306e\u5165\u308c\u66ff\u3048\u3067\u9006\u6570\u306b\uff09<\/p>\n

  8. \n$\\log_{x^p}z=\\displaystyle\\frac{\\log_x z}{\\log_x x^p}=\\displaystyle\\frac{\\log_x z}{p\\log_x x}=\\displaystyle\\frac{\\log_x z}{p\\cdot 1}=\\displaystyle\\frac{1}{p}\\log_x z=\\log_x z^\\frac{1}{p}$\uff08\u5e95\u306e$p$\u4e57\u306f\u771f\u6570\u306e$1\/p$\u4e57\uff09
    \n\u3000\u3000$\\left(y=\\log_{x^p}z\\Leftrightarrow (x^p)^y=(x^y)^p=z\\Rightarrow x^y=\\sqrt[p]{z}=z^{\\frac{1}{p}}\\Leftrightarrow y=\\log_x{z^{\\frac{1}{p}}} \\right)$<\/p>\n
  9. \n$\\log_{x^p}z^p=\\displaystyle\\frac{\\log_a z^p}{\\log_a x^p}=\\displaystyle\\frac{\\cancel{p}\\log_a z}{\\cancel{p}\\log_a x}=\\displaystyle\\frac{\\log_a z}{\\log_a x}=\\log_x z$\uff08\u5e95\u3068\u771f\u6570\u306e$p$\u4e57\u306f\u76f8\u6bba\uff09
    \n\u3000\u3000$\\left(\u4e0a\u5f0f\u3067\u5e95\u3092x\u3068\u3057\u3066\u3001\\log_{x^p}z^p=\\displaystyle\\frac{\\log_x z^p}{\\log_x x^p}=\\displaystyle\\frac{\\cancel{p}\\log_x z}{\\cancel{p}\\log_x x}=\\displaystyle\\frac{\\log_x z}{1}=\\log_x z\\right)$
    \n\u3000\u3000$\\left(z^p=(x^p)^y=(x^y)^p \\Rightarrow \\sqrt[p]{z^p}=\\sqrt[p]{(x^y)^p} \\Rightarrow z=x^y \\right)$<\/p>\n
  10. \n$\\log_a b\\cdot \\log_b c=\\displaystyle\\frac{\\cancel{\\log_x b}}{\\log_x a}\\displaystyle\\frac{\\log_x c}{\\cancel{\\log_x b}}=\\displaystyle\\frac{\\log_x c}{\\log_x a}=\\log_a c$\uff08\u771f\u6570\u3068\u5e95\u306e\u9023\u9396\u3067\u76f8\u6bba\uff09
    \n\u3000\u3000$\\left(\u4e0a\u5f0f\u3067\u5e95\u3092a\u3068\u3057\u3066\u3001\\log_a b\\cdot \\log_b c=\\cancel{\\log_a b}\\displaystyle\\frac{\\log_a c}{\\cancel{\\log_a b}}=\\log_a c\\right)$<\/p>\n
  11. \n\u5e38\u7528\u5bfe\u6570\u304b\u3089\u81ea\u7136\u5bfe\u6570\u3078\u306e\u5909\u63db $\\ln z=\\displaystyle\\frac{\\log_{10}z}{\\log_{10}{\\rm e}}=\\displaystyle\\frac{\\log_{10}z}{\\log_{10}2.718\\cdots}=\\displaystyle\\frac{\\log_{10}z}{0.434\\cdots}\\fallingdotseq 2.3\\log_{10}z$\n<\/ol>\n<\/div>\n