---
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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
-\begin{equation}\begin{split}
+\begin{equation*}\begin{split}
a^m \times a^n & = a^{m+n} \\
- a^m \div a^n & = a^{m-n}4 \\
+ 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}
-\end{split}\end{equation}
-
-## Fractional indices
-
-$$^n\sqrt{x}=x^{1/n}$$
-
-## Logarithms
-
-$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
-
-## Using logs to solve index eq's
-
-Used for equations without common base exponent
-
-Or change base:
-$$\log_b c = {{\log_a c} \over {\log_a b}}$$
-
-If $a<1, \quad \log_{b} a < 0$ (flip inequality operator)
-
-## Exponential functions
-
-$e^x$ - natural exponential function
-
-
-$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
+ {({a \over b})}^m & = {a^m \over b^m} \\
+ ^n\sqrt{x} &=x^{1/n}
+\end{split}\end{equation*}
## Logarithm laws
-\begin{equation}\begin{split}
+\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
-\end{split}\end{equation}
-
+ \log_a1 = 0 & \text{ and } \log_aa = 1 \\
+ \log_b c &= {{\log_a c} \over {\log_a b}}
+\end{split}\end{equation*}
## Inverse functions
For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
-$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$
+$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax$$
-## Euler's number
-
-$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
+## Exponentials
-## Literal equations
+$$e^x \quad \text{natural exponential function}$$
-_Literal equation_ - no numerical solutions
+$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
-## Exponential and logarithmic modelling
+## Modelling
$$A = A_0 e^{kt}$$
-where
-$A_0$ is initial value
-$t$ is time taken
-$k$ is a constant
-For continuous growth, $k > 0$
-For continuous decay, $k < 0$
+- $A_0$ is initial value
+- $t$ is time taken
+- $k$ is a constant
+- For continuous growth, $k > 0$
+- For continuous decay, $k < 0$
+
+\columnbreak
-## Graphing expomnential functions
+## Graphing exponential functions
$$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$
-- **$y$-intercept** at $(0, A \cdot a^{-kb}+c)$
+- **$y$-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 $|A|$ 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 $y=x$)
+$\log_e x$ 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 $\mathbb{R}^+$
+- **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 $|A|$ from $x$-axis
- dilation of factor $1 \over k$ from $y$-axis
+{#id .class width=30%}
+
+## Finding equations
+
+\colorbox{cas}{On CAS:} {#id .class width=75px}