Exponential and logarithmic functions

Builds on

Required prerequisites, all the way down: 24

You will learn to

• Extend exponents to negative and rational values and justify the definitions from the laws of exponents.
• Solve exponential equations with logarithms and apply the laws of logarithms.
• Model growth and decay with exponential functions and compare exponential with polynomial growth.

Definition

• For a > 0, a^(1/n) is the positive n-th root of a, and a^(m/n) = (a^(1/n))^m.
• For a > 0 with a ≠ 1 and x > 0, log_a x is the number y with a^y = x.
• e is the limit of (1 + 1/n)ⁿ as n grows without bound; e ≈ 2.71828.

Theorem

• log_a(xy) = log_a x + log_a y and log_a(x^r) = r log_a x.
• For a > 1, aˣ eventually exceeds every polynomial in x.
• For a > 0 with a ≠ 1, x ↦ aˣ is a bijection from ℝ onto (0, ∞) whose inverse is log_a.
• e^(x + y) = eˣ · eʸ: the exponential function turns addition into multiplication.

Common misconceptions

• Writing log(x + y) = log x + log y.
• Thinking a⁻ⁿ is negative.
• Confusing exponential growth (multiplying by the same factor) with linear growth (adding the same amount).

Leads to

Optional discoveries

• Structure-preserving maps: homomorphisms and isomorphisms exp is an isomorphism from (ℝ, +) to (ℝ>0, ×), with inverse ln: it translates one structure exactly into the other.

Learning pathways

• Applied mathematics

Further reading

• Silvanus P. Thompson, Calculus Made Easy (1910)