---
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-<!-- columns: 2 -->
-graphics: yes
-tables: yes
+geometry: a4paper, margin=2cm
+columns: 2
author: Andrew Lorimer
-classoption: twocolumn
-header-includes: \pagenumbering{gobble}
+header-includes:
+- \usepackage{fancyhdr}
+- \usepackage{setspace}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+- \usepackage{graphicx}
+- \usepackage{tabularx}
+- \usepackage[dvipsnames]{xcolor}
---
-# Exponential and Index Functions
+\pagenumbering{gobble}
+\setstretch{1.5}
+\definecolor{cas}{HTML}{e6f0fe}
+
+# Exponentials & Logarithms
## Index laws
-$a^m \times a^n = a^{m+n}$
-$a^m \div a^n = a^{m-n}4$
-$(a^m)^n = a^{_mn}$
-$(ab)^m = a^m b^m$
-${({a \over b})}^m = {a^m \over b^m}$
+\begin{equation*}\begin{split}
+ a^m \times a^n & = a^{m+n} \\
+ a^m \div a^n & = a^{m-n} \\
+ (a^m)^n & = a^{_mn} \\
+ (ab)^m & = a^m b^m \\
+ {({a \over b})}^m & = {a^m \over b^m} \\
+ ^n\sqrt{x} &=x^{1/n}
+\end{split}\end{equation*}
-## Fractional indices
+## Logarithm laws
-$^n\sqrt{x}=x^{1/n}$
+\begin{equation*}\begin{split}
+ \log_a(mn) & = \log_am + \log_an \\
+ \log_a({m \over n}) & = \log_am - \log_a \\
+ \log_a(m^p) & = p\log_am \\
+ \log_a(m^{-1}) & = -\log_am \\
+ \log_a1 = 0 & \text{ and } \log_aa = 1 \\
+ \log_b c &= {{\log_a c} \over {\log_a b}}
+\end{split}\end{equation*}
-## Logarithms
+## Inverse functions
-$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
+For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
-## Using logs to solve index eq's
+$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax$$
-Used for equations without common base exponent
+## Exponentials
-Or change base:
-$$\log_b c = {{\log_a c} \over {\log_a b}}$$
+$$e^x \quad \text{natural exponential function}$$
-If $a<1, \quad \log_{b} a < 0$ (flip inequality operator)
+$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
-## Exponential functions
+## Modelling
-$e^x$ - natural exponential function
+$$A = A_0 e^{kt}$$
+- $A_0$ is initial value
+- $t$ is time taken
+- $k$ is a constant
+- For continuous growth, $k > 0$
+- For continuous decay, $k < 0$
-$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
+\columnbreak
-## Logarithm laws
+## Graphing exponential functions
-$\log_a(mn) = \log_am + \log_an$
-$\log_a({m \over n}) = \log_am - \log_an$
-$\log_a(m^p) = p\log_am$
-$\log_a(m^{-1}) = -\log_am$
-$\log_a1 = 0$ and $\log_aa = 1$
+$$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$
-## Inverse functions
+- **$y$-intercept** at $(0, A \cdot a^{-kb}+c)$ as $x \rightarrow \infty$
+- **horizontal asymptote** at $y=c$
+- **domain** is $\mathbb{R}$
+- **range** is $(c, \infty)$
+- dilation of factor $|A|$ from $x$-axis
+- dilation of factor $1 \over k$ from $y$-axis
+
+{#id .class width=30%}
+
+## Graphing logarithmic functions
+
+$\log_e x$ is the inverse of $e^x$ (reflection across $y=x$)
+
+$$f(x)=A \log_a k(x-b) + c$$
+
+where
+
+- **domain** is $(b, \infty)$
+- **range** is $\mathbb{R}$
+- **vertical asymptote** at $x=b$
+- $y$-intercept exists if $b<0$
+- dilation of factor $|A|$ from $x$-axis
+- dilation of factor $1 \over k$ from $y$-axis
+
+{#id .class width=30%}
+
+## Finding equations
-Inverse of $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$ is $f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$
+\colorbox{cas}{On CAS:} {#id .class width=75px}