M5.2.5 For the adult
The number e and natural logs
Is there a best base for exponentials and logarithms, and what is the number e?
What this chapter teaches
The number e is the value that (1 + 1/n)ⁿ approaches as n grows, about 2.71828. The natural logarithm ln x is the log to the base e, and it undoes eˣ. The graphs of eˣ and ln x are mirror images in y = x. The gradient of eˣ is eˣ, a fact proved later in differentiation.
Three questions to ask
There is no one right wording. Listen for the idea.
- What happens to (1 + 1/n)ⁿ as n gets bigger, and what do we call the number it settles on?
- How can you solve eˣ = 5, and what answer should a calculator give?
- Why does ln (−3) have no value?
After the chapter, the learner can
- Find e as the value of (1 + 1/n)ⁿ for large n
- Sketch y = eˣ and y = ln x as mirror images
- Solve e²ˣ = 7 using natural logs
What they copy into their notebook
e is about 2.71828, the limit of (1 + 1/n)ⁿ. ln x is the log to base e and undoes eˣ. ln (eˣ) = x. The graphs are mirror images in y = x.