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   1---
   2geometry: margin=1.5cm
   3<!-- columns: 2 -->
   4graphics: yes
   5tables: yes
   6author: Andrew Lorimer
   7classoption: twocolumn
   8header-includes: \pagenumbering{gobble}
   9---
  10
  11# Exponential and Index Functions
  12
  13## Index laws
  14
  15\begin{equation}\begin{split}
  16  a^m \times a^n & = a^{m+n} \\
  17  a^m \div a^n & = a^{m-n}4 \\
  18  (a^m)^n & = a^{_mn} \\
  19  (ab)^m & = a^m b^m \\
  20  {({a \over b})}^m & = {a^m \over b^m}
  21\end{split}\end{equation}
  22
  23## Fractional indices
  24
  25$$^n\sqrt{x}=x^{1/n}$$
  26
  27## Logarithms
  28
  29$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
  30
  31## Using logs to solve index eq's
  32
  33Used for equations without common base exponent
  34
  35Or change base:  
  36$$\log_b c = {{\log_a c} \over {\log_a b}}$$
  37
  38If $a<1, \quad \log_{b} a < 0$ (flip inequality operator)
  39
  40## Exponential functions
  41
  42$e^x$ - natural exponential function
  43
  44$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
  45
  46## Logarithm laws
  47
  48\begin{equation}\begin{split}
  49  \log_a(mn) & = \log_am + \log_an \\
  50  \log_a({m \over n}) & = \log_am - \log_a \\
  51  \log_a(m^p) & = p\log_am \\
  52  \log_a(m^{-1}) & = -\log_am \\
  53  \log_a1 = 0 & \text{ and } \log_aa = 1
  54\end{split}\end{equation}
  55
  56
  57## Inverse functions
  58
  59For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
  60
  61$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$
  62
  63## Euler's number
  64
  65$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
  66
  67## Exponential and logarithmic modelling
  68
  69$$A = A_0 e^{kt}$$
  70
  71where  
  72$A_0$ is initial value  
  73$t$ is time taken  
  74$k$ is a constant  
  75For continuous growth, $k > 0$  
  76For continuous decay, $k < 0$
  77
  78## Graphing exponential functions
  79
  80$$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$
  81
  82- **$y$-intercept** at $(0, A \cdot a^{-kb}+c)$ as $x \rightarrow \infty$
  83- **horizontal asymptote** at $y=c$
  84- **domain** is $\mathbb{R}$
  85- **range** is $(c, \infty)$
  86- dilation of factor $A$ from $x$-axis
  87- dilation of factor $1 \over k$ from $y$-axis
  88
  89![](graphics/exponential-graphs.png){#id .class width=30%} 
  90
  91## Graphing logarithmic functions
  92
  93$\log_e x$ is the inverse of $e^x$ (reflection across $y=x$)
  94
  95$$f(x)=A \log_a k(x-b) + c$$
  96
  97where
  98
  99- **domain** is $(b, \infty)$
 100- **range** is $\mathbb{R}$
 101- **vertical asymptote** at $x=b$
 102- $y$-intercept exists if $b<0$
 103- dilation of factor $A$ from $x$-axis
 104- dilation of factor $1 \over k$ from $y$-axis
 105
 106![](graphics/log-graphs.png){#id .class width=30%} 
 107
 108## Finding equations
 109
 110Solve simultaneous equations on CAS: ![](graphics/cas-simultaneous.png){#id .class width=75px}