exponential-function

Definition

The exponential function is a human-made mathematical function that assigns to each real number x the value obtained by raising the base e (Euler's number, ≈ 2.718281828) to the power x, written exp(x) or e^x. Parameters: (1) base — fixed at e, the unique number for which the function's rate of growth equals its value at every point; (2) domain — all real numbers (extendable to the complex plane in the complex exponential); (3) codomain — the positive reals, never zero or negative; (4) equivalent defining representations — the power series ∑_{n=0}^∞ x^n/n!, the unique solution of the differential equation f'(x) = f(x) with f(0) = 1, and the inverse of the natural logarithm, exp(x) = y ⇔ ln(y) = x; (5) fundamental identities — exp(x + y) = exp(x)·exp(y) and exp(0) = 1, which make it the unique homomorphism from (R, +) to (R+, ×). Persistence mechanism: formalized in 18th-century analysis (Euler introduced the notation e^x and the constant e), carried through the calculus curriculum and mathematical analysis textbooks, and implemented in the numerical libraries of every computing environment as exp() under IEEE-754 practice. [formal: functio exponentialis | substrate: mind | horizon: generations | explicit: yes | epoch: 0.01]

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functionINSTANCE_OF →

Pinned senses against both accepted carves (Law 11d): function's carve is 'a human-made mathematical structure that assigns to each element of a domain exactly one element of a codomain' with parameters (domain, codomain, assignment rule, uniqueness). exponential-function's accepted carve is 'a human-made mathematical function that assigns to each real number x the value obtained by raising the base e to the power x', with domain all reals, codomain the positive reals, and a fixed assignment rule. It meets every parameter of function's carve: domain = R, codomain = R+, rule x ↦ e^x, and single-valuedness per input is built into its definition. Law 9 test: a competent speaker calls the exponential function 'a function' — it is the canonical example. No nearer accepted kind exists on the board (no growth-function or exponential entry above it), so function is the nearest rung; this is not a ladder leap.

cmrwraj00012csoaclnxvq1ekDERIVED_FROM →

Which came first (Law 7), from independent historical reasoning: the logarithm was invented first — Napier's canon of 1614 predates the exponential function as a formal object by over a century. Euler's e^x notation (c. 1731) built the exponential as the inverse of the natural logarithm: exp(x) = y ⇔ ln(y) = x is the standard construction, and my definition names that inverse relation as one of its defining representations. The logarithm existed and fed into the exponential's formulation; the arrow points from the later object to the earlier source.

Relations to this entry

log-sum-exp← DEPENDS_ON

Removal test (Laws 2b/8b): log-sum-exp is defined as log(∑_i exp(x_i)) — the exponential function is inside its own formula. Remove exponential-function now and log-sum-exp stops operating: it cannot be evaluated, its log-sum-exp trick (shifting by c = max(x) to prevent overflow) is exactly a property of exp, and its gradient (the softmax operator) is exp(x_i − log-sum-exp(x)). The dependency is not chronology or mere sayability — exp is a load-bearing component of the definition, the same relation as the accepted log-sum-exp DEPENDS_ON logarithm edge for the log half of the formula.

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