{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exponential functions and logarithms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import math\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exponential functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is **e**? It is simply a number (known as Euler's number):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.718281828459045"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "math.e"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**e** is a significant number, because it is the base rate of growth shared by all continually growing processes.\n",
    "\n",
    "For example, if I have **10 dollars**, and it grows 100% in 1 year (compounding continuously), I end up with **10\\*e^1 dollars**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "27.18281828459045"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 100% growth for 1 year\n",
    "10 * np.exp(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "73.890560989306508"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 100% growth for 2 years\n",
    "10 * np.exp(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Side note: When e is raised to a power, it is known as **the exponential function**. Technically, any number can be the base, and it would still be known as **an exponential function** (such as 2^5). But in our context, the base of the exponential function is assumed to be e.\n",
    "\n",
    "Anyway, what if I only have 20% growth instead of 100% growth?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12.214027581601698"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 20% growth for 1 year\n",
    "10 * np.exp(0.20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "14.918246976412703"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 20% growth for 2 years\n",
    "10 * np.exp(0.20 * 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logarithms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the **(natural) logarithm**? It gives you the time needed to reach a certain level of growth. For example, if I want growth by a factor of 2.718, it will take me 1 unit of time (assuming a 100% growth rate):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.99989631572895199"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# time needed to grow 1 unit to 2.718 units\n",
    "np.log(2.718)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If I want growth by a factor of 7.389, it will take me 2 units of time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.9999924078065106"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# time needed to grow 1 unit to 7.389 units\n",
    "np.log(7.389)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If I want growth by a factor of 1, it will take me 0 units of time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# time needed to grow 1 unit to 1 unit\n",
    "np.log(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If I want growth by a factor of 0.5, it will take me -0.693 units of time (which is like looking back in time):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.69314718055994529"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# time needed to grow 1 unit to 0.5 units\n",
    "np.log(0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Connecting the concepts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, the exponential function and the natural logarithm are **inverses** of one another:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.log(np.exp(5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.9999999999999991"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.exp(np.log(5))"
   ]
  }
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