{ "id": "exponential-function", "version": "3b851b935ff9", "status": "published", "name": "Exponential Function Calculator", "question": "What is my exponential function?", "summary": "Evaluates f(x) = a·bˣ, or finds the exponential function through two points, with the percent growth or decay rate, the continuous rate k = ln b, and the doubling time or half-life.", "category": "math", "subcategory": "algebra", "url": "https://www.acalculator.org/math/exponential-function-calculator", "markdown": "https://www.acalculator.org/math/exponential-function-calculator.md", "kind": "function", "method": "f(x) = a·bˣ; from two points, b = (y₂ ÷ y₁)^(1 ÷ (x₂ − x₁)) and a = y₁ ÷ b^x₁; rate = (b − 1) × 100%; k = ln b; doubling time or half-life = ln 2 ÷ |ln b|.", "assumptions": [ "a ≠ 0, b > 0 and b ≠ 1 (OpenStax’s definition of an exponential function).", "Typed numbers are exact decimals; whole powers are exact, roots and fractional powers are decimal numbers." ], "inputs": { "$schema": "https://json-schema.org/draft/2020-12/schema", "type": "object", "properties": { "mode": { "title": "I know", "description": "a and b of f(x) = a·bˣ, or two points the function passes through.", "type": "string", "enum": [ "ab", "points" ] }, "a": { "title": "a (initial value)", "description": "f(0), the value where x = 0. It may not be 0.", "type": "number", "minimum": -1000000000, "maximum": 1000000000 }, "b": { "title": "b (base)", "description": "The growth factor per step of x: more than 0, and not 1.", "type": "number", "minimum": 0, "maximum": 1000000000 }, "x1": { "title": "x₁", "description": "The x of the first point.", "type": "number", "minimum": -1000000000, "maximum": 1000000000 }, "y1": { "title": "y₁", "description": "The y of the first point. Not 0.", "type": "number", "minimum": -1000000000, "maximum": 1000000000 }, "x2": { "title": "x₂", "description": "The x of the second point. Not the same as x₁.", "type": "number", "minimum": -1000000000, "maximum": 1000000000 }, "y2": { "title": "y₂", "description": "The y of the second point: not 0, and the same sign as y₁.", "type": "number", "minimum": -1000000000, "maximum": 1000000000 }, "x": { "title": "x", "description": "Where to evaluate the function.", "type": "number", "minimum": -10000, "maximum": 10000 } } }, "outputs": { "fx": { "label": "f(x)", "description": "a·bˣ at the x you typed.", "format": "number" }, "equation": { "label": "Function", "description": "f(x) = a·bˣ with a and b to 10 significant figures.", "format": "text" }, "a": { "label": "a (initial value)", "description": "The value of f at x = 0.", "format": "number" }, "b": { "label": "b (growth factor)", "description": "The factor f grows by each time x goes up by 1.", "format": "number" }, "rate": { "label": "Rate per step", "description": "(b − 1) × 100: positive for growth, negative for decay.", "format": "percent" }, "kind": { "label": "Growth or decay", "description": "Growth when b > 1, decay when 0 < b < 1.", "format": "text" }, "k": { "label": "Continuous rate k", "description": "k = ln b, so f(x) = a·e^(kx).", "format": "number" }, "doubling": { "label": "Doubling time or half-life", "description": "ln 2 ÷ |ln b|: the change in x that doubles (growth) or halves (decay) f.", "format": "number" } }, "defaultAnswer": { "inputs": { "mode": "ab", "a": 100, "b": 1.05, "x1": -2, "y1": 6, "x2": 2, "y2": 1, "x": 10 }, "outputs": { "fx": 162.88946267774415, "equation": "f(x) = 100·1.05ˣ", "a": 100, "b": 1.05, "rate": 5.000000000000004, "kind": "Exponential growth", "k": 0.04879016416943205, "doubling": 14.206699082890461 }, "text": "f(x) = 100·1.05ˣ; at your x, f(x) = 162.8894627." }, "examples": [ { "given": { "mode": "points", "x1": -2, "y1": 6, "x2": 2, "y2": 1, "x": 0 }, "expect": { "a": 2.449489742783178, "b": 0.6389431042462724, "fx": 2.449489742783178, "kind": "Exponential decay" }, "source": "OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions, Example 5 (f(x) = 2.4492(0.6389)ˣ through (−2, 6) and (2, 1)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions; full precision: b = (1/6)^(1/4), a = 6b²", "tolerance": 1e-12 }, { "given": { "mode": "ab", "a": 100, "b": 1.05, "x": 10 }, "expect": { "fx": 162.88946267774415, "rate": 5, "kind": "Exponential growth", "doubling": 14.206699082890461 }, "source": "OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions (f(x) = abˣ, b > 1 is growth), https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions; hand calculation in content.mdx: 100 × 1.05¹⁰ = 162.889462677744140625 exactly; ln 2 ÷ ln 1.05 = 14.2067" }, { "given": { "mode": "points", "x1": 0, "y1": 3, "x2": 1, "y2": 12, "x": 3 }, "expect": { "a": 3, "b": 4, "fx": 192, "rate": 300 }, "source": "OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions (writing f(x) = abˣ from two points), https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions; hand calculation in content.mdx: a = 3, b = 4, f(3) = 3 × 64 = 192" }, { "given": { "mode": "ab", "a": 80, "b": 0.5, "x": -2 }, "expect": { "fx": 320, "doubling": 1, "rate": -50 }, "source": "OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions (0 < b < 1 is decay), https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions; hand calculation in content.mdx: 80 × 0.5⁻² = 320; half-life ln 2 ÷ ln 2 = 1" } ], "sources": [ "OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions (f(x) = abˣ with a nonzero and b positive, b ≠ 1; Example 5: through (−2, 6) and (2, 1), f(x) = 2.4492(0.6389)ˣ). https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions (retrieved 2026-10-02)" ], "related": [ "exponential-growth", "exponent", "log", "growth-rate" ], "changelog": [] }