Exponential function

In Year One and Year Two we learned to compute powers $a^x$ for natural, integer and rational exponents. In Year Three we take the decisive step: we accept that $a^x$ is well defined for every real number $x$, and that the properties of powers continue to hold. Thus the exponential function $\exp_a(x)=a^x$ is born, one of the most important functions in all of mathematics. The chapter studies its graph and its properties (domain, image, monotonicity, asymptote), the methods for solving exponential equations and inequalities (same base, substitution, change of direction), and finally the many real phenomena it describes: radioactive decay and carbon-14 dating, compound interest and continuous compounding, the absorption of light, atmospheric pressure and the cooling of bodies.

Sections

• From powers to the exponential function
• Graph and properties
• Exponential equations
• Exponential inequalities
• Models of exponential growth

Exercises

• All the worked exercises of the chapter