Exponential, and logarithmic functions are pivotal mathematical concepts that play central roles in advanced mathematics. Unfortunately, these are also concepts that give students serious difficulty. This report describe a theory of how students acquire an understanding of these functions by prescribing a set of mental constructions that a student can make to develop his or her understanding of these concepts. We analyze students' understanding of these concepts within the context of our theory. Our main result was that while all of the students in the study could compute exponents in simple cases, few students could reason about the process of exponentiation. Thus, according to our theory, these students' knowledge of exponential and logarithmic function will be limited. We conclude by describing instructional activities based on our theoretical analysis designed to foster students' understanding of these concepts. (Author)

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