## Motivation

### Two approaches

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.

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