This book presents a clarifying account of the ideas associated with logarithms, powers, and exponentials. Students who have been puzzled by presentations in textbooks, math teachers, and writers of instructional material will find new, simpler, and more visual ways to understand and talk about the ideas. There are roughly four innovations here.

First, primacy of the log scale (a.k.a. "multiplicative number line") is urged. The log scale is indispensable for apprehending the tiny, numerous, fast entities that form our world. It is widely used in scientific and engineering practice; it is even found on sites presenting demographic data. Surprisingly, it is a hitherto unacknowledged tool for thinking about ratios, powers, and logarithms.

Second, new conceptions of exponents and logarithms are given. From Newton until our day, the rather awkward notion of a real exponent acting via a limit of powers with rational exponents has held sway. Using the geometric grounding of the log scale, this book aims to convert the reader's mind to persistently viewing an exponent as a scaling of a directed segment on the log scale. A logarithm is then a corresponding ratio. Once grasped, these twin conceptions have a compelling simplicity and visualizability. They actually coincide with modern definitions and computations, unlike the traditional conception based on limits of rational powers.

Third, a new, mathematical, and elementary account of the naturalness of the natural logarithm and exponential is given. This account provides intuition for various expressions of these functions and a viable story of the significance of the number e. It leads to a statement of a mathematical "law of natural growth": add here, multiply there, in the simplest way possible on a microscopic level.

Fourth, all of this is extended seamlessly into the complex number domain. The log scale is embedded in the "log scroll"—the Riemann surface of the logarithm. The idea of exponents as scalings adapts without a hitch and clarifies complex powers. Gauss's contour integral definition of the natural logarithm admits ready interpretation in accord with the previously stated law of natural growth.

This book begins with the log scale and ends with the log scroll. The former is understandable to eighth graders; the latter is usually not considered until one reaches upper-level undergraduate mathematics. In the end one can see that each part of the development is all of a piece. The traditional development of this subject matter is not really coherent. No wonder it continues to baffle many students and that people continue to question the meaning of "natural" exponential and the number e. This book brings out the clarity and elegance that one suspects the subject should actually possess.