In this module, we will introduce two new functions \(e^x\) and \(\log_e x\). We will do this in two different ways.
The first approach develops the topic in an investigatory fashion, starting from the question: `What is the derivative of \(2^x\)?' However, as we proceed, we will point out some shortcomings of this approach.
Alternatively, we can begin from a definition of \(\log_e x\) as an integral, and then define \(e^x\) as its inverse. The story is then told in a completely different order.
The first approach is probably easier for most students to understand, but the second approach is more concise and rigorous.
In general, when telling a mathematical story, there are various goals such as elegance, rigour, practicality, generality and understandability. Sometimes these goals conflict, and we have to compromise. Sometimes developing a subject in the most logically concise way does not make for easy reading. As with any other subject, learning mathematics from multiple perspectives leads to a deeper and more critical understanding.