Section 1.5: Exponential Functions

Exponential Functions

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.

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. a^x = a^yif and only ifx = y.

3. For x ≠ 0, a^x = b^xif and only ifa = b.

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 = e^x.

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, ∞).

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 = 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.