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+geometry: margin=1.5cm
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graphics: yes
tables: yes
## 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}4 \\
+ (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
-$^n\sqrt{x}=x^{1/n}$
+$$^n\sqrt{x}=x^{1/n}$$
## Logarithms
$e^x$ - natural exponential function
-
$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
## Logarithm laws
-$\log_a(mn) = \log_am + \log_an$
-$\log_a({m \over n}) = \log_am - \log_an$
-$\log_a(m^p) = p\log_am$
-$\log_a(m^{-1}) = -\log_am$
-$\log_a1 = 0$ and $\log_aa = 1$
+\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}
## Inverse functions
-Inverse of $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$ is $f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$
+For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
+
+$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$
## Euler's number
$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
-## Literal equations
-
-_Literal equation_ - no numerical solutions
-
## Exponential and logarithmic modelling
$$A = A_0 e^{kt}$$
For continuous growth, $k > 0$
For continuous decay, $k < 0$
-## Graphing expomnential functions
+## Graphing exponential functions
-$$f(x)=Aa^{k(x-b)} + c, \quad \vert a > 1$$
+$$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$
-- **$y$-intercept** at $(0, {{1+c} \over {a^b}})$
+- **$y$-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 $y=x$)
+$\log_e x$ 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 (reflection across $x$-axis when $A < 0$)
-- dilation of factor $1 \over k$ from $y$-axis (reflection across $y$-axis when $k < 0$)
+- dilation of factor $A$ from $x$-axis
+- dilation of factor $1 \over k$ from $y$-axis
+
+{#id .class width=30%}
+
+## Finding equations
+Solve simultaneous equations on CAS: {#id .class width=75px}