The algebra of exponents are grouped into those with the same base...

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

...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 \(e\)

\(e^x\) is an exponential 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.