# Section 1.5: Exponential Functions

## Exponential Functions

Exponential functions are of the form: f(x) = bx, 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.

### Properties of Exponential Functions

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

1. Exponent laws:
 axay = ax + y ax/ay = ax - y (ax)y = axy (ab)x = axbx (a/b)x = ax/bx
2. ax = ayif and only ifx = y.

3. For x ≠ 0, ax = bxif and only ifa = b.

b = 2

## Base e Exponential Functions

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 y = ex.

The exponential function with base 1/e is defined by y = e-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, ∞).

x = 1

## Simple, Compound, and Continuous Compound Interest

• 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 = Pert

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.