Chapter 4: Exponential and Logarithmic Functions

Chapter 4: Exponential and Logarithmic Functions

4.1 Exponential Functions

Objectives

• Evaluate exponential functions.

• Graph exponential functions.

• Evaluate functions with base $e$.

• Use compound interest formulas.

Definition of the Exponential Function

• An exponential function $f$ with base $b$ is defined as: $f(x) = a \times b^x$, where:

  • $b > 0$ and $b \neq 1$.

  • $a$ is the initial amount.

  • $b$ is the growth ($b > 1$) or decay ($0 < b < 1$) factor.

  • $x$ represents time intervals.

• If $a=1$, the function simplifies to $f(x) = b^x$.

• Real-world applications include bacterial growth/decay, population changes, and compound interest.

Example: Evaluating an Exponential Function

• The function $f(x) = 42.2(1.56)^x$ models average spending at a mall after $x$ hours.

• For $x=3$ hours, the average amount spent is $160.

Graphing an Exponential Function

• Graphing considers the nature of exponential functions.

• Exponential Growth: When $b > 1$, graphs curve upwards (e.g., $f(x) = 2^x$).

• Exponential Decay: When $0 < b < 1$, graphs curve downwards (e.g., $f(x) = 2^{-x}$).

Characteristics of Exponential Functions

• Exponential functions are one-to-one.

• Graphs are affected by translations, reflections, and transformations.

Compound Interest

• Interest is calculated on the principal and accumulated interest.

• Formulas depend on compounding intervals; continuous compounding uses base $e$.

Example of Historical Context in Compound Interest

• A debt from 1461, compounded at 4% until 1996, could have grown to approximately $290 billion.