ABSTRACT
The sixth chapter looks into the power law, exponential, and logarithmic functions. We know that in order to raise a number to a positive integer power we just need to multiply it by itself the required number of times, but how do we rigorously define negative, rational, and real exponents? This chapter carefully extends exponentiation from natural exponents to all rational numbers. The properties of the power law function form the foundation for defining the exponential function and the logarithm. Many previously proved theorems about functions in general come into play to prove, for example, that the logarithmic function exists. All standard properties of these functions, often presented “as is” in high school textbooks are proved. These proofs delineate the applicability of exponentiation, which is important for applications. For example, some properties of exponentiation do not apply to the case of raising a negative number to a rational power, and neglecting this fact often leads to errors, which is demonstrated in this chapter.
