Through an account of the history of exponential functions as presented in traditional calculus textbooks, I present my observations and remarks on the spiral development of the concept, and my concerns about the general presentations of the subject. In this article I emphasize how the different arrangements and sequencing of the subjects required for the development of the concept of the logarithm would lead to different definitions. A slow process that requires gradual maturity and readiness precedes the formal foundation of the concept. One statement can be considered a definition according to one arrangement, and a theorem according to another arrangement. This study is primarily addressed to teachers, or to mathematics education students in a content-based mathematics education class.