-# Index laws
+---
+geometry: margin=2cm
+<!-- columns: 2 -->
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
+header-includes: \pagenumbering{gobble}
+---
+
+# Exponential and Index Functions
+
+## Index laws
$a^m \times a^n = a^{m+n}$
$a^m \div a^n = a^{m-n}4$
$(ab)^m = a^m b^m$
${({a \over b})}^m = {a^m \over b^m}$
-# Fractional indices
+## 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
+
+
+$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
+
+## 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 $\log_aa = 1$
+
+
+## Inverse functions
+
+Inverse of $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$ is $f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$
+
+## Euler's number
+
+$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
+
+## Literal equations
+
+_Literal equation_ - no numerical solutions
+
+## Exponential and logarithmic modelling
+
+$$A = A_0 e^{kt}$$
+
+where
+$A_0$ is initial value
+$t$ is time taken
+$k$ is a constant
+For continuous growth, $k > 0$
+For continuous decay, $k < 0$
+m
+## Graphing expomnential functions
+
+$$f(x)=Aa^{k(x-b)} + c, \quad \vert a > 1$$
+
+- **$y$-intercept** at $(0, {{1+c} \over {a^b}})$
+- **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
+
-$^n\sqrt{x}=x^{1/n}$
\ No newline at end of file