Natural growth — money, populations, isotopes, cooling coffee — is exponential. Calculating with it demands fluency in both eˣ and its inverse ln x, and the strange-but-wonderful fact that eˣ is its own derivative. This chapter builds that fluency from the ground up.
1 · Exponentials and logarithms as inverses
Exponential laws: b^(x+y) = bˣbʸ, b^(x−y) = bˣ/bʸ, (bˣ)^y = b^(xy), b⁰ = 1. The logarithm undoes the exponent: log_b x is the power to which b must be raised to produce x. They cancel each other both ways — log_b(bˣ) = x and b^(log_b x) = x — and ln x is simply log_e x. Domains matter: y = ln x exists only for x > 0.
Three log rules do most of the work: the product rule log_b(MN) = log_b M + log_b N, the quotient rule log_b(M/N) = log_b M − log_b N, and the power rule log_b(Mʳ) = r·log_b M. Change-of-base (log_b M = ln M/ln b) converts anything into ln, your calculator's native tongue.
2 · Birth of e: compound interest
Invest P at annual rate r and compound n times per year: A = P(1 + r/n)^(nt). Compounding yearly, quarterly, monthly… the factor (1 + r/n)^n creeps upward toward a ceiling. That ceiling defines the number e: e = lim_{n→∞}(1 + 1/n)^n ≈ 2.71828, equivalently e = lim_{x→0}(1 + x)^(1/x). e is irrational and transcendental, and it is the base of natural growth — the only base whose curve rises at the rate of its own height.
3 · Differentiating eˣ, aˣ, ln x, log_a x
From the limit definition of e one shows (eˣ)′ = eˣ: the slope equals the height, at every point, exactly. Everything else follows by the chain or inverse-function rule: (aˣ)′ = aˣ·ln a, (ln x)′ = 1/x for x > 0, and (log_a x)′ = 1/(x ln a). Finally, with the power rule extended from integers to all real exponents, (xᵃ)′ = a x^(a−1) — now valid for fractional and irrational a, proved via xᵃ = e^(a ln x).
4 · Limits involving e
Infinite compounding is the launchpad: (1 + 1/n)^n → e as n → ∞, and more generally (1 + k/n)^n → eᵏ. The mirrored form lim_{x→0}(1 + x)^(1/x) = e shows up in the definition of eˣ's derivative. Know the asymptotics cold: eˣ → ∞ as x → ∞, eˣ → 0 (positive) as x → −∞, ln x → ∞ as x → ∞, and ln x → −∞ as x → 0⁺. So eˣ has a horizontal asymptote at y = 0 while ln x has a vertical one at x = 0.
Differentiate y = xˣ (x > 0) — neither the power rule nor the exponential rule fits alone.
Step 1. Take ln: ln y = x ln x.
Step 2. Differentiate implicitly: y′/y = ln x + x·(1/x) = ln x + 1.
Step 3. Multiply by y: y′ = xˣ(ln x + 1).
The same recipe handles x^(sin x), giving x^(sin x)·(cos x·ln x + sin x/x).
5 · Logarithmic differentiation
The trick generalises: whenever the variable sits in both base and exponent (xˣ, x^(sin x)) or in sprawling products, quotients and powers, slap ln on both sides first. The ln power rule flattens exponents into multipliers, and implicit differentiation finishes. This is also the cleanest proof that (xᵃ)′ = a x^(a−1) for arbitrary real a.
6 · Exponential growth and decay
If a quantity's rate of change is proportional to itself, dy/dt = ky, then y = y₀e^(kt). Positive k models growth (bank balances, bacteria), negative k decay (isotopes, drug clearance). From one datum find k, then predict. Doubling time: t₂ = ln 2/k; half-life: t½ = ln 2/|k|. Example: a 200 g isotope with half-life 4 days retains 200·(½)^(t/4) grams after t days — after 12 days, 25 g remain.
7 · Hyperbolic functions
Combine eˣ and e⁻ˣ symmetrically: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x/cosh x. They mirror circular trig: cosh²x − sinh²x = 1, the derivatives are sinh′ = cosh, cosh′ = sinh (no minus sign!), and tanh′ = sech²x. sinh is odd, cosh is even with minimum value 1 — the catenary shape of a hanging chain.
eˣ is not differentiated by the power rule: d/dx eˣ = eˣ, not x·e^(x−1). The two rules apply to different signatures — exponential base xᵃ versus base-with-exponent-variable aˣ — and mixing them is the classic silent error.
log_b(MN) = log_b M + log_b N
log_b(M/N) = log_b M − log_b N
log_b(Mʳ) = r log_b M
e = lim_{n→∞} (1 + 1/n)^n
(eˣ)′ = eˣ, (aˣ)′ = aˣ ln a
(ln x)′ = 1/x, (log_a x)′ = 1/(x ln a)
y = y₀e^(kt), t½ = ln 2/|k|
sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, cosh²x − sinh²x = 1
Common mistakes
- Applying the power rule to eˣ — the derivative is eˣ, never x·eˣ⁻¹.
- Taking ln of a non-positive number without checking the domain; y = ln x needs x > 0.
- Mis-signing the chain rule in (aˣ)′ = aˣ ln a — that extra ln a is mandatory for a ≠ e.
- Confusing log_b M with log_b N in inversion: log₅25 = 2, not 5.
- Writing the decay model with k positive — decay needs k < 0 inside y = y₀e^(kt).
- Forgetting the implicit step in logarithmic differentiation: always multiply by y at the end.