eˣ 指数函数计算器（求值、反解 x、两点拟合）Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points)

使用说明与常见问题How-to & FAQ

「指数函数计算器（求值、反解 x、两点拟合）」是什么？What is the Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points) tool?

处理 y = a·b^(kx)：给定 x 求 y，给定 y 反解 x，或由两个点求出 a 和 b；同时给出增长/衰减率、翻倍时间或半衰期，以及等价的 a·e^(rx) 形式。本地处理。Work with y = a·b^(kx): find y from x, solve for x from y, or find a and b from two points, with the growth or decay rate, the doubling time or half-life, and the equivalent a·e^(rx) form. Processed locally.

如何使用「指数函数计算器（求值、反解 x、两点拟合）」？How do I use the Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points) tool?

在「指数函数计算器（求值、反解 x、两点拟合）」中，填写「计算」、「初始值 a」、「底数 b（e 可写 e）」、「指数系数 k」、「x」等参数，结果会随输入实时计算并显示在下方。In the Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points): Fill in "Calculate", "Initial value a", "Base b (write e for e)", "Exponent factor k", "x"; results are recalculated instantly as you type.

怎么由两个点求指数函数？How do I find an exponential through two points?

设 y = a·b^x，两式相除得 b = (y₂/y₁)^(1/(x₂ − x₁))，再代回得 a = y₁ ÷ b^x₁。例如 (0, 5) 和 (3, 40)：b = 8^(1/3) = 2，a = 5，所以 y = 5·2^x。With y = a·b^x, dividing the two equations gives b = (y₂/y₁)^(1/(x₂ − x₁)), and then a = y₁ ÷ b^x₁. For (0, 5) and (3, 40): b = 8^(1/3) = 2 and a = 5, so y = 5·2^x.

怎么反解 x？How is x solved from y?

取对数：x = ln(y/a) ÷ (k·ln b)。y 和 a 必须同号，否则函数取不到这个值。Take logs: x = ln(y/a) ÷ (k·ln b). y and a must have the same sign, otherwise the function never reaches that value.

翻倍时间和半衰期怎么来的？Where do the doubling time and half-life come from?

写成 y = a·e^(rx)，r = k·ln b。r > 0 时翻倍需要 ln 2 ÷ r，r < 0 时减半需要 ln 2 ÷ |r|，单位和 x 一样。Written as y = a·e^(rx) with r = k·ln b, the doubling time is ln 2 ÷ r when r > 0 and the half-life is ln 2 ÷ |r| when r < 0, in the same units as x.

能举个计算实例吗？Can you show a worked example?

例如在「指数函数计算器（求值、反解 x、两点拟合）」中填 计算 = 由 x 求 y，初始值 a = 1，底数 b（e 可写 e） = 2，指数系数 k = 1，x = 3，结果区显示：y = a·b^(kx)：8；连续增长率 r（y = a·e^(rx)）：0.69314718；每单位 x 的变化：100%；翻倍所需 x：1For example, entering Calculate = y from x, Initial value a = 1, Base b (write e for e) = 2, Exponent factor k = 1, x = 3 in the Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points) gives: y = a·b^(kx): 8; Continuous rate r (y = a·e^(rx)): 0.69314718; Change per unit of x: 100%; x to double: 1

哪些输入会被检查？Which inputs are validated?

「指数函数计算器（求值、反解 x、两点拟合）」会检查这些输入——「初始值 a」、「x」、「y」：所选方式用到时必须填写；「指数系数 k」、「点 1 的 x₁」、「点 1 的 y₁」、「点 2 的 x₂」、「点 2 的 y₂」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points) checks its inputs — "Initial value a", "x", "y": required when the chosen option uses it; "Exponent factor k", "Point 1 x₁", "Point 1 y₁", "Point 2 x₂", "Point 2 y₂": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「指数函数计算器（求值、反解 x、两点拟合）」的设置：「计算」可选 由 x 求 y、由 y 反解 x、由两个点求函数（默认 由 x 求 y）；「初始值 a」默认 1；「底数 b（e 可写 e）」默认 2；「指数系数 k」默认 1；「x」默认 3；「y」默认 8；「点 1 的 x₁」默认 0；「点 1 的 y₁」默认 5；「点 2 的 x₂」默认 3；「点 2 的 y₂」默认 40。每次生成结果都按当前设置处理。Settings in the Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points) — "Calculate": y from x, x from y, Function through two points (default y from x); "Initial value a" default 1; "Base b (write e for e)" default 2; "Exponent factor k" default 1; "x" default 3; "y" default 8; "Point 1 x₁" default 0; "Point 1 y₁" default 5; "Point 2 x₂" default 3; "Point 2 y₂" default 40. The result always uses the current settings.

计算精度如何？How precise are the results?

「指数函数计算器（求值、反解 x、两点拟合）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「指数函数计算器（求值、反解 x、两点拟合）」免费吗？需要注册吗？Is the Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points) free? Do I need an account?

「指数函数计算器（求值、反解 x、两点拟合）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Exponential Function Calculator (Evaluate, Solve for x, Fit Two Points) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.