Exponential Function Calculator — f(x) = a × bˣ
Evaluate the exponential function f(x) = a × bˣ at any x value. Enter the initial value a, the base b, and x to see the result, growth or decay rate, doubling time or half-life, and an interactive plot of the curve.
1 × 2^5 = 32
- 1
Evaluate bˣ
2 ˣ (x = 5) = 32The base raised to the power x. - 2
Multiply by coefficient a
1 × 32 = 32
How does this calculator work?
f(x) = a × bˣ: a is the starting value (f(0)), b is the per-step multiplier. If b > 1: exponential growth with doubling time = ln(2)/ln(b). If 0 < b < 1: exponential decay with half-life = ln(2)/|ln(b)|. Enter a, b, and x to evaluate the function and plot the curve.
Formula
How this is calculated
An exponential function f(x) = a × bˣ has two parameters: the coefficient a, which equals f(0) (the function value at x = 0, sometimes called the initial amount), and the base b, which is the multiplicative factor applied per unit increase in x. If b > 1 the function grows — each step multiplies the current value by b, so the rate of change is proportional to the current value, not constant. If 0 < b < 1 the function decays — each step multiplies by a fraction, shrinking f toward zero. The case b = 1 gives a constant, and b ≤ 0 is undefined for non-integer x.
The doubling time (when b > 1) or half-life (when b < 1) is the interval of x over which f doubles or halves. Solving a × bᵗ = 2a gives t = ln(2)/ln(b), which is always positive regardless of whether b > 1 or b < 1 (taking |ln(b)|). For compound interest with b = 1 + r, this matches the rule-of-72 approximation closely. For radioactive decay, b = e^(−λ) and half-life = ln(2)/λ.
The chart plots f over the range x ± 5 around the entered x, marking the evaluated point. For very large x with b >> 1, the function can overflow IEEE-754 doubles (~10¹⁵); the plot clips such values automatically and the calculator shows a warning if no finite result is reachable.
Frequently asked questions
A function is exponential if its rate of change is proportional to its current value — mathematically, if df/dx = k × f(x). This means the function multiplies by a constant factor b for each unit increase in x, rather than adding a constant (which would give linear growth). The constant factor is what makes growth "explode" or decay "never quite reach zero".
Set a × bᵗ = 2a and solve: bᵗ = 2, so t = log_b(2) = ln(2)/ln(b). For b = 2 (doubling each step), t = 1 unit. For b = 1.07 (7% annual growth), t = ln(2)/ln(1.07) ≈ 10.24 years — close to the rule-of-72 estimate of 72/7 ≈ 10.3.
Setting a = 1 and b = e ≈ 2.71828 gives the natural exponential f(x) = eˣ, the unique function equal to its own derivative. It underlies continuous compounding (e^(rt)), population dynamics, and radioactive decay. Enter b = 2.71828 (or more precisely 2.718281828) in this calculator to use the natural base.
TG we-Calculate Editorial Team. (2026). Exponential Function Calculator — f(x) = a × bˣ [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/exponential-function-calculator
TG we-Calculate Editorial Team. "Exponential Function Calculator — f(x) = a × bˣ." TG we-Calculate. 2026. https://we-calculate.com/calculator/exponential-function-calculator.
TG we-Calculate Editorial Team, "Exponential Function Calculator — f(x) = a × bˣ," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/exponential-function-calculator
@misc{wecalculate_exponential_function_calculator, title = {Exponential Function Calculator — f(x) = a × bˣ}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/exponential-function-calculator}}, year = {2026}, note = {TG we-Calculate} }
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