CHAPTER 1

1.1 Introduction

Exponential and logarithms functions are important concepts that play crucial roles in college mathematics courses, including calculus, differential equations, and complex analysis. The purpose of this study is to describe a theory of how students might develop their understanding of these topics and to analyze understanding of these concepts within the context of this theory, their application in real life phenomena and discussing with their model.

One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range (Lial & Miller, 1975). The exponential function is not to be confused with the polynomial functions, such as x2. One way to recognize the difference between the two functions is by the name of the function. Exponential functions are called so because the variable lies within the exponent of the function (Allendoerfer, Oakley, & Kerr, 1977). These functions are often recognized by the fact that their rate of growth is proportional to their value (Bogley & Robson, 1999). This concept of exponential growth has been around much longer than at the dawn of calculus. Evidence of this dates back almost 4,000 years ago on a Mesopotamian clay tablet, which is now on display at the Louvre. The question translated from the stone slab simply asks, “How long will it take for a sum of money to double if invested at 20 percent interest rate compounded annually?”(Aleff, 2005).

The intensity of exponential function study began in the late 17th and early 18th centuries, when great mathematicians, such as Jacob Bernoulli, Leonard Euler, and Isaac Newton began to delve into the depths of exponential and logarithmic functions. Their insights brought about the creation of modern calculus, which in turn gave...

1.1 Introduction

Exponential and logarithms functions are important concepts that play crucial roles in college mathematics courses, including calculus, differential equations, and complex analysis. The purpose of this study is to describe a theory of how students might develop their understanding of these topics and to analyze understanding of these concepts within the context of this theory, their application in real life phenomena and discussing with their model.

One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range (Lial & Miller, 1975). The exponential function is not to be confused with the polynomial functions, such as x2. One way to recognize the difference between the two functions is by the name of the function. Exponential functions are called so because the variable lies within the exponent of the function (Allendoerfer, Oakley, & Kerr, 1977). These functions are often recognized by the fact that their rate of growth is proportional to their value (Bogley & Robson, 1999). This concept of exponential growth has been around much longer than at the dawn of calculus. Evidence of this dates back almost 4,000 years ago on a Mesopotamian clay tablet, which is now on display at the Louvre. The question translated from the stone slab simply asks, “How long will it take for a sum of money to double if invested at 20 percent interest rate compounded annually?”(Aleff, 2005).

The intensity of exponential function study began in the late 17th and early 18th centuries, when great mathematicians, such as Jacob Bernoulli, Leonard Euler, and Isaac Newton began to delve into the depths of exponential and logarithmic functions. Their insights brought about the creation of modern calculus, which in turn gave...