Exponential Functions, Logarithms, and the Number e

The number $e \approx 2.71828$ is not a curiosity — it is the base forced by the internal logic of calculus. Every continuous-time finance model uses $e$ because continuous compounding is the natural limit of discrete compounding, and because $e^x$ is the unique function equal to its own derivative. This note derives $e$ from the compound-interest limit, establishes the algebra of exponential and logarithmic functions, and develops the inverse-function machinery that converts between them. The continuous compounding formula $A = Pe^{rt}$ is the central output; understanding its derivation is the correct foundation for every yield-curve, option-pricing, and stochastic-rate model you will encounter.

• sketch $y = a^x$ for $a > 0$, $a \neq 1$, and state its domain, range, and asymptote;
• derive the compound-interest limit $\lim_{n\to\infty}(1 + r/n)^n = e^r$ and state the value of $e$;
• state and apply the four laws of exponents;
• determine whether a function is one-to-one using the Horizontal Line Test;
• find the inverse of a one-to-one function algebraically and graphically;
• convert between exponential and logarithmic form using $\log_a x = y \Leftrightarrow a^y = x$;
• apply the three laws of logarithms to expand or condense expressions;
• use the change-of-base formula $\log_a x = \ln x / \ln a$.

1. Exponential functions
2. The number e and the compound-interest limit
3. The number e more precisely
4. Inverse functions and logarithms
5. Growth comparison: exponential vs power vs logarithm