1---
2geometry: margin=2cm
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
45$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
46
47## Logarithm laws
48
49\begin{equation}\begin{split}
50 \log_a(mn) & = \log_am + \log_an \\
51 \log_a({m \over n}) & = \log_am - \log_a \\
52 \log_a(m^p) & = p\log_am \\
53 \log_a(m^{-1}) & = -\log_am \\
54 \log_a1 = 0 & \text{ and } \log_aa = 1
55\end{split}\end{equation}
56
57
58## Inverse functions
59
60For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
61
62$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$
63
64## Euler's number
65
66$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
67
68## Literal equations
69
70_Literal equation_ - no numerical solutions
71
72## Exponential and logarithmic modelling
73
74$$A = A_0 e^{kt}$$
75
76where
77$A_0$ is initial value
78$t$ is time taken
79$k$ is a constant
80For continuous growth, $k > 0$
81For continuous decay, $k < 0$
82
83## Graphing exponential functions
84
85$$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$
86
87- **$y$-intercept** at $(0, A \cdot a^{-kb}+c)$ as $x \rightarrow \infty$
88- **horizontal asymptote** at $y=c$
89- **domain** is $\mathbb{R}$
90- **range** is $(c, \infty)$
91- dilation of factor $A$ from $x$-axis
92- dilation of factor $1 \over k$ from $y$-axis
93
94## Graphing logarithmic functions
95
96$log_e x$ is the inverse of $e^x$ (reflection across $y=x$)
97
98$$f(x)=A \log_a k(x-b) + c$$
99
100where
101
102- **domain** is $(b, \infty)$
103- **range** is $\mathbb{R}$
104- **vertical asymptote** at $x=b$
105- $y$-intercept exists if $b<0$
106- dilation of factor $A$ from $x$-axis
107- dilation of factor $1 \over k$ from $y$-axis
108
109## Finding equations
110
111Solve simultaneous equations on CAS: ![](graphics/cas-simultaneous.png){#id .class width=75px}