• $A^n \cdot A^m = A^{n+m}$
• $\frac {A^n}{A^m} = A^{n-m}$
• $(A^n)^m = A^{n \cdot m}$
  

...and those with the same exponent.

• $A^n \cdot B^n = (A \cdot B)^n$
• $\frac {A^n}{B^n} = \left (\frac {A}{B}\right )^n$
  

These lead to equalities such as

• $A^0 = 1$
• $A^1 = A$
• $A^{-n} = \frac {1}{A^n}$

Our definition of function includes a rule that says each domain number is in exactly one pair. A One-to-one function satisfies the reverse. Each range number is in exactly one pair. Exponential functions are one-to-one functions.

In other words, each function value occurs exactly once in an exponential function. This is handy information when solving equations.

If you know that $A^r = A^t$, then $r=t$ follows.

$\blacktriangleright$ This is an important and useful rule.

e

We were introduced to $e$ earlier.

You know the number $\pi$ (pronounced ”pie”). The definition of $\pi$ is kind of weird. For any circle, $\pi$ is the ratio of the circumference to the diameter.

$\pi$ is approximately $3.1415926$.

There is another number that comes up quite frequently when dealing with natural growth. This number called “e” and its symbol is $e$.

$e$ is approximately $2.718281828$.

The definition of $e$ is even weirder.

Consider the function $e(x) = \left (1 + \frac {1}{x}\right )^x$

This function is an increasing function, but it increases at a smaller and smaller rate. The graph starts to level off and approaches a horizontal asymptote. This asymptote is at a height of $e$. And that is the definition of $e$

$$e = \lim _{x \to \infty } \left (1 + \frac {1}{x}\right )^x$$

$e^x$ is an exponetial function, which means it is a one-to-one function.

$e^x$ is an exponetial function, which means it is a one-to-one function, which means that it reaches every positive number, exactly once.