2018-04-03-2-2.ipynb

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    "### $$y=\\log_{a}(x)$$ の逆関数を説明できる\n",
    "- No: 2\n",
    "- Sub No: 2\n",
    "- 難易度: ★\n",
    "- カテゴリ名: 統計数理基礎\n",
    "- 必須スキル: ◯\n",
    "\n",
    "\n",
    "\n",
    "### 逆関数\n",
    "- $$y=f(x)$$ で表される式を $$x=g(y)$$ にしたもの\n",
    "- $$f(x)$$ 逆関数は $$f^{-1}(x)$$ と書く\n",
    "- $$y=f(x)$$ なら $$x=f^{-1}(y)$$\n",
    "- 直線 $$y=x$$ に関して対称になる\n",
    "\n",
    "\n",
    "### 逆関数の例\n",
    "\n",
    "$$f(x)=3x+1$$\n",
    "\n",
    "$$f^{-1}(x)=\\frac{x-1}{3}$$\n",
    "\n",
    "$$f(2)=7,f^{-1}(7)=2$$\n",
    "\n",
    "\n",
    "\n",
    "### log の逆関数\n",
    "\n",
    "$$f(x)=log_{a}(x)$$\n",
    "\n",
    "$$f^{-1}(x)=a^{x}$$\n",
    "\n",
    "※ a の部分は 底(てい) と呼ばれ、1以外の正の実数\n",
    "\n",
    "\n",
    "\n",
    "### Juliaで関数をプロットしてみる\n",
    "\n",
    "```julia\n",
    "f(x) = log(x) # 底(a) が ネイピア数e(2.7182...) の場合\n",
    "g(x) = exp(x) # e^x\n",
    "```\n",
    "\n",
    "\n",
    "```julia\n",
    "using Gadfly # グラフの描画\n",
    "\n",
    "# 描画用に範囲を決める\n",
    "log_range = 0.007:0.001:5.0 # 0.007から0.001刻みで5.0までの範囲を表す\n",
    "exp_range = -5.0:0.001:1.6\n",
    "x_range = -5.0:0.001:5.0\n",
    "\n",
    "# plotする\n",
    "Gadfly.plot(\n",
    "    layer(x = log_range, y = f(log_range), Geom.line, Theme(default_color=color(\"red\")) ),\n",
    "    layer(x = exp_range, y = g(exp_range), Geom.line, Theme(default_color=color(\"blue\")) ),\n",
    "    layer(x = x_range, y = x_range, Geom.line, Theme(default_color=color(\"black\")) ),\n",
    "    Guide.manual_color_key(\"式\",[\"log\", \"exp\"], [\"red\", \"blue\"])\n",
    ")\n",
    "```\n",
    "\n",
    "\n",
    "![](/images/log-exp.jpg)\n",
    "\n",
    "\n",
    "### 数値確認\n",
    "\n",
    "```julia\n",
    "f(1.0) # => 0.0\n",
    "g(0.0) # => 1.0\n",
    "f(0.1) # => -2.3025850929940455\n",
    "g(-2.3025850929940455) # => 0.10000000000000002\n",
    "```"
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