Exponential and Logarithmic Functions

Exponential Functions

Exponential functions are a fundamental class of functions in algebra, characterized by a constant base raised to a variable exponent. They are widely used to model growth and decay in various real-world contexts, such as population growth, radioactive decay, and financial investments.

• Definition: An exponential function with base b is defined as f(x) = b^x, where b is a positive constant other than 1 (b > 0 and b \neq 1), and x is any real number.

• Domain: All real numbers.

• Range: All positive real numbers (f(x) > 0).

• Horizontal Asymptote: The line y = 0 is a horizontal asymptote for all exponential functions of this form.

Evaluating Exponential Functions

To evaluate an exponential function at a given value of x, substitute the value into the function and compute the result.

• Example: If f(x) = 2^x, then f(3) = 2^3 = 8.

• Application: If f(x) = 80 \cdot 2^{0.5x} models the average amount spent at a shopping mall after x hours, then after 3 hours: (Rounded to the nearest dollar, $226$.)

Graphing Exponential Functions

Graphing exponential functions involves plotting points for various values of x and connecting them with a smooth curve. The graph always passes through (0,1) for f(x) = b^x and increases rapidly for b > 1.

• Key Features:

  • Passes through (0,1)

  • Horizontal asymptote at y = 0

  • Rapid growth for positive x if b > 1

Transformations of Exponential Functions

Exponential function graphs can be shifted, reflected, or stretched/compressed by modifying the function's formula. Common transformations include horizontal and vertical shifts.

• Horizontal Shift: f(x) = b^{x-h} shifts the graph h units to the right.

• Vertical Shift: f(x) = b^x + k shifts the graph k units up.

• Reflection: f(x) = b^{-x} reflects the graph across the y-axis.

• Example: To graph f(x) = 2^{x-1}, shift the graph of f(x) = 2^x one unit to the right.

Characteristics of Exponential Functions

• Continuous and smooth curves

• One-to-one functions (pass the horizontal line test)

• Always positive outputs for real-valued exponents

• Rapid growth or decay depending on the base

The Natural Base e

The number e is an important mathematical constant, approximately equal to 2.718281828. It is the base of the natural exponential function and arises naturally in many contexts involving continuous growth or decay.

• Definition:

• Natural Exponential Function: f(x) = e^x

• e is irrational (its decimal expansion never terminates or repeats)

Evaluating Functions with Base e

To evaluate a function with base e, substitute the value of x and use a calculator for the exponentiation.

• Example: If f(x) = 1200e^{0.04x} models the gray wolf population x years after 1978, then for 2017 (x = 39): (Population is approximately 5712 wolves.)

Compound Interest Formulas

Exponential functions are used to model compound interest, where interest is earned on both the initial principal and the accumulated interest.

• For n compounding periods per year: Where:

  • A = final amount

  • P = principal (initial amount)

  • r = annual interest rate (decimal)

  • n = number of compounding periods per year

  • t = number of years

• For continuous compounding:

Example: Quarterly Compounding

• Problem: annual interest, compounded quarterly, for 5 years.

• Solution:

Example: Continuous Compounding

• Problem: annual interest, compounded continuously, for 5 years.

• Solution: