Suppose that a := limx→∞ (1 + 1/x)x exists and is greater than 1 (see Example 4.22). Assume that ax: R → (0, ∞) is onto, continuous, strictly increasing, and satisfies axay = ax+y and (ax)y = axy for all x,y ∈ R (see Exercise 3.3.11). Let L(x) denote the inverse function of ax.
a) Prove that t L(l + 1/t) → 1 as t → ∞.
b) Prove that (ah - 1)/h → 1 as h → 0.
c) Prove that ax is differentiable on R and (ax)' = ax for all ∈ R.
d) Prove that L'(x) = l/x for all x > 0.
[A is the natural base e and L(JC) is the natural logarithm log*.]