---
geometry: margin=2cm
<!-- columns: 2 -->
graphics: yes
tables: yes
author: Andrew Lorimer
classoption: twocolumn
header-includes: \pagenumbering{gobble}
---

# Exponential and Index Functions

## Index laws

$a^m \times a^n = a^{m+n}$  
$a^m \div a^n = a^{m-n}4$  
$(a^m)^n = a^{_mn}$  
$(ab)^m = a^m b^m$  
${({a \over b})}^m = {a^m \over b^m}$

## 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