Was ‘e’ Found or Invented?

Students often make mistakes when manipulating exponentials logs in their A level Maths exams.

Perhaps this is because exponential functions and their inverses – logarithmic functions – are misunderstood as functions? If you think about it, there is a very big difference between y=2^x and y=x^2, which helps explain why the algebraic manipulation of exponentials and polynomials are so very different.

Your A level Maths exam will definitely have questions involving exponentials. You must be confident with exponential growth & decay, solving exponential equations and differentiation & integration of functions involving exponentials and logs.

In the 18th century, Euler was trying to solve a problem set by Bernoulli, and he came across a very special number. If you’ve ever wondered why it’s called ‘e’ (you probably haven’t) it’s because it’s Euler’s constant (e for Euler) – because he found it. Is ‘found’ the right word? Did e exist already and Euler found it or did Euler invent it to explain what he was seeing when he tried to solve problems with compound interest?  Welcome to the philosophy of maths!

We’ve learned a lot about e since the 18th century thanks to the hard work of a lot of mathematicians and physicists.  It turns out that as well as being fundamental to compound interest, e is also a vital component of the way we model waves such as light waves, sound waves and quantum waves.  

In your A level Maths course you will learn to solve exponential equations and logarithmic equations involving e^x and its inverse function, ln(x). The algebra required is a bit more tricky than you’ve got used to at GCSE because you need to use the laws of logs very carefully.

Law 1) ln(ab)=ln(a)+ln(b)

Law 2) ln(a/b)=ln(a)-ln(b)

Law 3) ln(xn)=nln(x)

You’ll also learn how exponentials are used in mathematical modelling across many areas of real life to model growth and decay but in my opinion, the A level Maths syllabus doesn’t do justice to this amazing and very beautiful number.

KEY POINTS: A level Maths + Natural Logs + e + Exponential Growth + Laws of Logs + Logs Equations

EXAM BOARDS: AQA, EDEXCEL, OCR, WJEC