Evaluate a × b^x, find doubling and halving times, and solve exponential equations for x with growth and decay explained.
Leave y blank to just evaluate f(x). A solution exists only when (y − c) / a is positive.
f(x)
How to use this tool
Type the coefficient a, the base b and the exponent x. The function is f(x) = a × b^x + c.
The headline is the value of the function at your x, with the per-step growth or decay rate alongside.
For growth, this is how many steps it takes to double; for decay, how many steps to halve. It is constant for a pure exponential.
Enter a target y to solve a × b^x + c = y. A solution exists only when (y − c) / a is positive, because b^x is always positive.
Formula & worked example
With a = 2, b = 3 and x = 2, f(x) = 2 × 3^2 = 18. The per-step growth is 200%, and the doubling time is ln 2 / ln 3 ≈ 0.631 steps.
Limitations — read this before you rely on it
The base must be positive and different from 1: b = 1 gives a flat line, not an exponential, and a negative base makes real powers undefined for most x. Very large exponents can overflow to infinity, in which case no finite answer exists.
Frequently asked questions
A function of the form f(x) = a × b^x, where the variable x is in the exponent. Each step multiplies the value by the same base b, so growth accelerates (or decay slows) rather than adding a fixed amount.
Linear growth adds the same amount each step; exponential growth multiplies by the same factor each step. After enough steps, exponential growth outruns any linear rate.
Divide ln 2 by ln b. For a base of 2 the doubling time is exactly 1 step; for a base of 1.1 it is ln 2 / ln 1.1 ≈ 7.27 steps.
Because b^x is always positive, a × b^x + c can never reach a y on the wrong side of c when a is positive (or the wrong side when a is negative). The calculator reports no solution in that case.
It shifts the whole curve up or down. It does not change the growth rate or the doubling time, but it does move the range of reachable y values.