activity_22_5_01_colab.ipynb

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    "kernelspec": {
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    "language_info": {
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      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.8.8"
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    "latex_envs": {
      "LaTeX_envs_menu_present": true,
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    "colab": {
      "name": "activity_22_5_01.ipynb",
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/28left/22jupyter/blob/main/activity_22_5_01_colab.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "TvzxvE3RYK75",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "79da32f6-071f-460d-c74f-b538a305ae86"
      },
      "source": [
        "!pip install git+https://github.com/28left/cyllene.git"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Collecting git+https://github.com/28left/cyllene.git\n",
            "  Cloning https://github.com/28left/cyllene.git to /tmp/pip-req-build-ytxfw2s1\n",
            "  Running command git clone -q https://github.com/28left/cyllene.git /tmp/pip-req-build-ytxfw2s1\n",
            "  Installing build dependencies ... \u001b[?25l\u001b[?25hdone\n",
            "  Getting requirements to build wheel ... \u001b[?25l\u001b[?25hdone\n",
            "    Preparing wheel metadata ... \u001b[?25l\u001b[?25hdone\n",
            "Requirement already satisfied: sympy in /usr/local/lib/python3.7/dist-packages (from cyllene==0.2) (1.7.1)\n",
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            "Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /usr/local/lib/python3.7/dist-packages (from matplotlib->cyllene==0.2) (2.4.7)\n",
            "Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.1->matplotlib->cyllene==0.2) (1.15.0)\n",
            "Building wheels for collected packages: cyllene\n",
            "  Building wheel for cyllene (PEP 517) ... \u001b[?25l\u001b[?25hdone\n",
            "  Created wheel for cyllene: filename=cyllene-0.2-cp37-none-any.whl size=27527 sha256=d21451174b843beac45ba734c3fe5e7ce9c26593d878e838a739b5e6ebb450c3\n",
            "  Stored in directory: /tmp/pip-ephem-wheel-cache-5q12d4bt/wheels/3e/18/73/4a45f074bb2820233eda690ac5f87b9c9f140d5442aa3fde8c\n",
            "Successfully built cyllene\n",
            "Installing collected packages: cyllene\n",
            "Successfully installed cyllene-0.2\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Wt6_o56MbPIR",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 667
        },
        "outputId": "aa4e75d2-963b-4d83-a49a-7e1421553252"
      },
      "source": [
        "!pip install \"ipython>=7\""
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Collecting ipython>=7\n",
            "\u001b[?25l  Downloading https://files.pythonhosted.org/packages/c9/b1/82cbe2b856386f44f37fdae54d9b425813bd86fe33385c9d658d64826098/ipython-7.22.0-py3-none-any.whl (785kB)\n",
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            "\u001b[?25hRequirement already satisfied: decorator in /usr/local/lib/python3.7/dist-packages (from ipython>=7) (4.4.2)\n",
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            "Collecting prompt-toolkit!=3.0.0,!=3.0.1,<3.1.0,>=2.0.0\n",
            "\u001b[?25l  Downloading https://files.pythonhosted.org/packages/eb/e6/4b4ca4fa94462d4560ba2f4e62e62108ab07be2e16a92e594e43b12d3300/prompt_toolkit-3.0.18-py3-none-any.whl (367kB)\n",
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            "\u001b[31mERROR: jupyter-console 5.2.0 has requirement prompt-toolkit<2.0.0,>=1.0.0, but you'll have prompt-toolkit 3.0.18 which is incompatible.\u001b[0m\n",
            "\u001b[31mERROR: google-colab 1.0.0 has requirement ipython~=5.5.0, but you'll have ipython 7.22.0 which is incompatible.\u001b[0m\n",
            "Installing collected packages: prompt-toolkit, ipython\n",
            "  Found existing installation: prompt-toolkit 1.0.18\n",
            "    Uninstalling prompt-toolkit-1.0.18:\n",
            "      Successfully uninstalled prompt-toolkit-1.0.18\n",
            "  Found existing installation: ipython 5.5.0\n",
            "    Uninstalling ipython-5.5.0:\n",
            "      Successfully uninstalled ipython-5.5.0\n",
            "Successfully installed ipython-7.22.0 prompt-toolkit-3.0.18\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
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              "pip_warning": {
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                  "IPython",
                  "prompt_toolkit"
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          "metadata": {
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    {
      "cell_type": "code",
      "metadata": {
        "id": "OMLVknKvZDA3"
      },
      "source": [
        "import cyllene\n",
        "from cyllene import *\n",
        "import sympy as sp\n",
        "from sympy import plot\n",
        "\n",
        "\n",
        "f = expression('x+2')\n",
        "\n",
        "g = expression('x^2')\n",
        "G_g = plot(g, (x, -3, 3), show=False)\n",
        "line = plot(4, (x, -5, 5), line_color='red',show=False)\n",
        "G_g.extend(line)\n",
        "\n",
        "G_h = plot(g, (x, 0, 3), show=False)\n",
        "line = plot(4, (x, 0, 4), line_color='red',show=False)\n",
        "G_h.extend(line)\n",
        "\n",
        "f_exp = function('2^x')\n",
        "P_1 = cyllene.pp.ExpressionProblem('1',\n",
        "                        'Compute the following values of $f(x) = 2^x$:', \n",
        "                        5,\n",
        "                        ['$f(-2)$', '$f(-1)$', '$f(0)$', '$f(1)$', '$f(2)$'],\n",
        "                        'numerical',\n",
        "                        [f_exp(-2), f_exp(-1), f_exp(0), f_exp(1), f_exp(2)]\n",
        "                       )\n",
        "ProbStack.add(P_1)\n",
        "\n",
        "P_2 = cyllene.pp.TrueFalse('2', \"The function $f(x) = 2^x$ is one-to-one\", True)\n",
        "ProbStack.add(P_2)\n",
        "\n",
        "f_log = function('log(x,2)')\n",
        "P_3 = cyllene.pp.ExpressionProblem('3',\n",
        "                        'Compute the following values of $f(x) = \\\\log_2(x)$:', \n",
        "                        3,\n",
        "                        ['$f(2)$', '$f(1)$', '$f(1/2)$'],\n",
        "                        'numerical',\n",
        "                        [f_log(2), f_log(1), sp.N(f_log(1/2))]\n",
        "                       )\n",
        "ProbStack.add(P_3)"
      ],
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "g483F0XKYJT5"
      },
      "source": [
        "# Activity 5 - Exponential and Logarithmic Functions"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GMtz6KeMYJT6"
      },
      "source": [
        "## Part 1 - Review of one-to-one functions and inverse functions"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "s64-jE33YJT6"
      },
      "source": [
        "### Introduction\n",
        "\n",
        "Recall what is meant by a one-to-one function. Given a function, we say that it is one-to-one if every element in the range of the function corresponds to only one element in the domain of a function.   That is, given a function $f(x)$, \n",
        "\n",
        "> if $f(x_1) = f(x_2)$, then $x_1 = x_2$.  \n",
        "\n",
        "(If the output was the same, then the input must have been the same.)\n",
        "\n",
        "Consider the following three functions, two of the functions are one-to-one functions; which one is NOT a one-to-one function?\n",
        "\n",
        "> 1. $f(x) = x + 2$ with domain $x \\in (-\\infty, \\infty )$\n",
        "> 2. $g(x) = x^2$ with domain $x \\in (-\\infty, \\infty )$\n",
        "> 3. $h(x) = x^2$ with domain $x \\in [0, \\infty)$\n",
        "\n",
        "In the past, you may have sketched the functions and employed the \"horizontal line test\" to determine whether or not the function was one-to-one. That is, if a horizontal line intersected the graph in more then one point, then the function was not one-to-one."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cXmnfo8kYJT6"
      },
      "source": [
        "Upon sketching the first function, $y=f(x)$, we see it is a line with slope $m=1$.\n",
        "\n",
        "Any horizontal line will cross this sketch in exactly one spot. \n",
        "\n",
        "In fact, given any value in the range of the function, for example $y=10$, we can calculate the corresponding $x$ value that gets mapped to it; that is, if $x+2=10$, then $x = 10-2 = 8$. [Note $f(8) = 10$]"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "id": "-yFcZ1kAYJT6",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 297
        },
        "outputId": "d995e18a-8741-4a91-a9e4-9f06f287dc5d"
      },
      "source": [
        "plot(f, (x,-12,12));"
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7OMBn2IpYJT7"
      },
      "source": [
        "On first glance, it may appear the the next two functions are the same; however, it is important to remember that associated with any functional definition is the domain of the function.  Even though $g(x)=x^2$ and $h(x)=x^2$, the domains are different, thus they are really two different functions.  The function $g(x)$ defined on all real numbers is NOT a one-to-one function. \n",
        "\n",
        "A counter example would be $g(-2) = (-2)^2 = 4$ and $g(2) = 2^2 = 4$.  Given an output, $y=4$, we have two inputs from the domain, $x_1= -2$ and $x_2=2$ that get mapped to $y=4$.  Sketch the parabola $y=x^2$ from $x=-3$ to $x=3$ and verify that the horizontal line, $y=4$, intersects the graph at two points, $(-2, 4)$ and $(2, 4)$.  Thus the function $g(x)$ as defined is NOT a one-to-one function.\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "id": "cuCCrs5sYJT7",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 296
        },
        "outputId": "2babaf2a-6a3f-4da3-824f-d748b6ab348e"
      },
      "source": [
        "G_g.show();"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "c7Kj9qL2YJT7"
      },
      "source": [
        "However, if you restrict your domain to just $x \\ge 0$, the \"sketch\" of the function will only be the right half of the parabola you sketched.  The horizontal line $y=4$ will only cross the graph in one place.  In fact, any horizontal line $y=k$ where $k>0$ will pass through the right half of parabola exactly once.  The third function is a one-to-one function."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "id": "HTrqXWc9YJT8",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 296
        },
        "outputId": "f8e26595-92a1-455a-8041-17d37d404971"
      },
      "source": [
        "G_h.show();"
      ],
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xIPglqBuYJT8"
      },
      "source": [
        "### The consequence of a one-to-one determination: The existence of an inverse function!"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xnVg-hYeYJT8"
      },
      "source": [
        "If a function is one-to-one, then an inverse *function* exists.  For $f(x) = x + 2$ with domain $x \\in (-\\infty, \\infty )$, we may write out the inverse function by writing $y = x + 2$ and solving for $x$, that is, $x = y-2$, which represents the inverse function, written $f^{-1}(y) = y-2$. Employing the forward function, $f(8) = 8+2 = 10$. Then, the inverse, $f^{-1}(10) = 10-2 = 8$, takes us back!\n",
        "\n",
        "For the function $h(x) = x^2$ with domain $x \\in [0, \\infty)$, we can do a similar process to determine the inverse function.  We let $y = x^2$ and solve for $x$ to get $x = \\sqrt{y}$ and $x = -\\sqrt{y}$. However, since we restricted the domain of $h(x)$ to be $x \\ge 0$, we need only consider the positive square root and $h^{-1}(y) = \\sqrt{y}$.  \n",
        "\n",
        "Notice of the second function from the Introduction, $g(x)$, since the domain was not resricted, the inverse process is not a \"functional\" process; for example, if $y=4$, going in reverse, there are two $x$-values associated with it, $x=-\\sqrt{4}=-2$ and $x=\\sqrt{4}=2$ (the horizontal line test).\n",
        "\n",
        "In short, given $y = f(x)$, then $x = f^{-1}(y)$.  Also note that the domain of the forward function $f(x)$ is the range of the inverse function, and the range of $f(x)$ is the domain of the inverse function."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_xL1X5yAYJT8"
      },
      "source": [
        "### A note on notation. \n",
        "\n",
        "Given $f(x) = x+2$, in lieu of writing $f^{-1}(y) = y-2$, since the variable $y$ represents the input and $y-2$ is the rule (what to do with the input), often the inverse is also written with the dummy variable $x$, that is, $f^{-1}(x) = x-2$.  However, the \"$x$\" in $f(x)$ and the \"$x$\" in $f^{-1}(x)$ represent two different quantities, most often with different units.  \n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1NNHZzHqYJT9"
      },
      "source": [
        "### Explore the outputs of a basic exponential function\n",
        "\n",
        "Consider the function $f(x) = 2^x$ defined on $(-\\infty, \\infty)$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aijWyGBFYJT9"
      },
      "source": [
        "## Exercise"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "scrolled": true,
        "id": "46h07hK4YJT9",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 244
        },
        "outputId": "4114142a-ff73-47a9-da1a-49b01c5bda4a"
      },
      "source": [
        "%problem 1"
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "### Problem 1",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "Compute the following values of $f(x) = 2^x$:",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "**(1)** &nbsp;&nbsp;$f(-2)$",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "**(2)** &nbsp;&nbsp;$f(-1)$",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "**(3)** &nbsp;&nbsp;$f(0)$",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "**(4)** &nbsp;&nbsp;$f(1)$",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "**(5)** &nbsp;&nbsp;$f(2)$",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "*Enter the answer(s) in the cell below.*",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "DeLYiD6BYJT9",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 175
        },
        "outputId": "738a1f8a-d9fe-4f6f-8690-a483d0d858d1"
      },
      "source": [
        "%%answer Problem 1\n",
        "(1): 1/4\n",
        "(2): 1/2\n",
        "(3): 1\n",
        "(4): 2\n",
        "(5): 4"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "You entered: <br>&#9989; &nbsp;**(1)** &nbsp;$\\displaystyle \\frac{1}{4}$ &nbsp;&nbsp; (*Correct*) <br>&#9989; &nbsp;**(2)** &nbsp;$\\displaystyle \\frac{1}{2}$ &nbsp;&nbsp; (*Correct*) <br>&#9989; &nbsp;**(3)** &nbsp;$\\displaystyle 1$ &nbsp;&nbsp; (*Correct*) <br>&#9989; &nbsp;**(4)** &nbsp;$\\displaystyle 2$ &nbsp;&nbsp; (*Correct*) <br>&#9989; &nbsp;**(5)** &nbsp;$\\displaystyle 4$ &nbsp;&nbsp; (*Correct*) <br>",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KwdoO_sXYJT-"
      },
      "source": [
        "## Exercise\n",
        "\n",
        "Below is a plot of the function $f(x) = 2^x$. Use the horizontal line test to determine if the function is one-to-one."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "id": "W6sTkJBKYJT-",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 412
        },
        "outputId": "0d497968-0acc-4302-8ac9-9face23e6666"
      },
      "source": [
        "plot(f_exp(x), (x,-3,3));\n",
        "%problem 2"
      ],
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "### Problem 2",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "**True or False?**",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "The function $f(x) = 2^x$ is one-to-one",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "*Enter `T` or `F` in the cell below.*",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-5ULj6XwYJT-",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 46
        },
        "outputId": "959ac579-5e85-470d-f05e-ab4e66c5064c"
      },
      "source": [
        "%%answer Problem 2\n",
        "T"
      ],
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "&#9989; &nbsp; **Correct!**",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Xxg-w6I8YJT_"
      },
      "source": [
        "Since the function $f(x) = 2^x$ is one-to-one, we know that a unique inverse function exists.  We can't just write $y = 2^x$ and \"solve for $x$\", we need new notation to depict this new function.\n",
        "\n",
        "> If $f(x) = 2^x$, then $f^{-1}(y) = \\log_2(y)$.\n",
        "\n",
        "In the graph above, consider the horizontal line through $y=4$ on the vertical axis and a vertical line through $x=2$ on the horizontal axis and note that it intersects the curve at the point $(2, 4)$.  This may be interpreted as $2^2=4$ or as $\\log_2(4) = 2$\n",
        "\n",
        "Using this idea, confirm the following values of $\\log_2(y)$: "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FPnXBxKqYJT_",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 177
        },
        "outputId": "7f080979-53f8-4a0a-867e-073c1d01bde3"
      },
      "source": [
        "%problem 3"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "### Problem 3",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "Compute the following values of $f(x) = \\log_2(x)$:",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "**(1)** $\\quad$$f(2)$",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "**(2)** $\\quad$$f(1)$",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "**(3)** $\\quad$$f(1/2)$",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "text/markdown": "*Enter the answer(s) in the cell below.*",
            "text/plain": [
              "<IPython.core.display.Markdown object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "289NhrobYJT_"
      },
      "source": [
        "### Closing Considerations\n",
        "\n",
        ">1. If $y  = b^x$, where wher the base $b>0$ and $b \\neq 1$, then $x = \\log_b (y)$; that is, given $y = f(x)$, then $x = f^{-1}(y)$.\n",
        "\n",
        ">2. Note that the domain (valid input values) of $f(x) = 2^x$ is $(-\\infty, \\infty)$, but the range (outputs) of $f(x) = 2^x > 0$.  \n",
        "This implies the domain of $f^{-1}(x) = \\log_2(x)$ is $x>0$ (See the note on notation above).  \n",
        "In general, the only valid input value into $\\log_b(x)$ is a value $x>0$ (e.g. the domain of the function $\\log_b(x)$ is $x>0$)\n",
        "\n",
        ">3. Since $2^0 = 1$, and in general $b^0 = 1$, we have $\\log_b(1) = 0$ regardless of base $b$."
      ]
    }
  ]
}