




Properties of Logarithmic Functions  Solving Exponential Equations
Each topic includes lecture notes with an interactive demonstration, a video, and testing content.


How to Teach with these Materials

Motivation 

A logarithmic spiral has the property that the rate at which the curve moves away from its origin increases with each rotation. Logarithmic spirals can be found in the shell of a nautilus, in weather patterns, and even in the structure of galaxies. Jakob Bernoulli was fascinated by this spiral, calling it spira mirabilis (miraculous spiral), and left instructions in his will to have one engraved on his tombstone, along with the phrase Eadem mutata resurgoâ€”"Although changed, I shall arise the same," possibly referring to the selfsimilar nature of the spiral. The engraver of his tombstone got the phrase right, but instead of a logarithmic spiral he carved an Archimedean spiral, which is what you get if you just coil up a rope on a flat surface  the radius of the spiral grows as a linear function of the rotation angle.










