By now we know Euler’s number \(\mathrm{e} =\mathrm{ e}^{1}\) quite well. In this chapter we define the exponential function \(\mathrm{e}^{x}\) for any x ∈ R, and its inverse the natural logarithmic function ln(x), for x > 0. (In the first section of the chapter we take a concise approach to the exponential function; in the second section we do things carefully.) These functions enable us to extend many of our previous results to allow for real exponents. For example, we obtain the Power Rule for real exponents, we extend Bernoulli’s Inequality, and we obtain a more strapping version of the AGM Inequality. We also meet the Logarithmic Mean, the Harmonic series and its close relatives the Alternating Harmonic series and p-series, and Euler’s constant \(\upgamma\).