I rather like Feynman's approach in the lecture Algebra from the Feynman Lectures https://www.feynmanlectures.caltech.edu/I_22.html

He covers the inverse of the exponential, Henry Briggs' log tables and goes on to e^ix = cos x + i sin x

The audio is also available https://www.feynmanlectures.caltech.edu/flptapes.html