Indices

Index Laws

$a^m\times a^n=a^{m+n}$ $a^m÷a^n=a^{m-n}$ $a^0=1$ $a^{-m}=\frac{1}{a^m}$ $(a^m{)}^n=a^{mn}$ $(ab{)}^n=a^n{b}^n$ $(\frac{a}{b}{)}^n=\frac{a^n}{b^n}$

Rational Indices

$a^{\frac{m}{n}}={}^n\sqrt{a^m}$ $(a^{\frac{1}{n}}{)}^n=a$

• In the previous sub-topic, we stated the index laws for $m$ and $n$ integers
• These laws are applicable for all rational indices

Standard form and scientific notation

• The number is expressed as a product of:
  • A number between 1 and 10
  • A power of 10

Graphs of exponential functions

$b\times a^{cx}+d$

• y intercept is at $d+1$
• b is dilation parallel to y axis
• $\frac{1}{c}$ is the dilation parallel to the x axis
• It never actually reaches 0 (forms an asymptote at y=0) $(\frac{1}{z}{)}^x=z^{-x}$

Solving Exponential Equations and Inequalities

• You literally just use mental math
• Teacher really went “just get good”
• ok uh
• if $a^x=a^y$, it is assumed that $x=y$
• One method for solving an exponential equation, without using a calculator, is first to express both sides of the equation as powers with the same base and then to equate the indices

The actual way

• you use logarithmic functions
• If $a^x=y$ $x=\frac{log(y)}{log(a)}$

Exponential Models and Applications

• In the following, we consider a variable $A$ that is subjective to exponential change
• Let $A$ be the quantity at time $t$
• Thus, $A=A_0{b}^t$, where $A_0$ is the initial quantity and $b$ is a positive constant
  • If