Exponential functions are of the form: f(x) = b^{x}, b > 0, b ≠ 1.

*b*is a positive constant (not equal to 1), called the**base**.- The
**domain**of an exponential function is the set of all real numbers. - The
**range**of an exponential function is the set of all positive real numbers.

Drag the slider to see the graph of the exponential function for different values of *b*.

For *a* and *b* positive, *a ≠ 1, b ≠ 1*, and *x* and *y* real,

- Exponent laws:
a ^{x}a^{y}= a^{x + y}a ^{x}/a^{y}= a^{x - y}(a ^{x})^{y}= a^{xy}(ab) ^{x}= a^{x}b^{x}(a/b) ^{x}= a^{x}/b^{x} a

^{x}= a^{y}if and only ifx = y.- For x ≠ 0, a
^{x}= b^{x}if and only ifa = b.

The number *e* can be approximated by (1 + 1/x)^{x} for large values of *x*.

Drag the slider to see the value of this expression for different values of *x*.

The exponential function with base *e* is defined by ^{x}

The exponential function with base *1/e* is defined by ^{-x}

- The
**domain**of these functions is the set of all real numbers, (-∞, ∞). - The
**range**of these functions is the set of all positive real numbers, (0, ∞).

- Simple interest is calculated using A = P(1 + rt)
- Compound interest is calculated using A = P(1 + r/m)
^{mt} - Continuous compound interest is calculated using A = Pe
^{rt}

where

*A*= amount owed*P*= amount borrowed (the "principal")*r*= annual interest rate (a decimal)*t*= time in years*m*= number of compounding periods per year.