How this calculator works
An exponential function multiplies a fixed scale by a positive base raised to x. The continuous form writes that base as e^c. Evaluation substitutes x, inversion divides out the scale then takes logarithms, and fitting uses the ratio of two non-zero values at distinct x coordinates. The shifts p and q move the curve horizontally and vertically. Subtract q before fitting or inversion; the adjusted point values must share a non-zero sign. Fitting returns a scaled form because both the scale and base are determined.
Formula
f(x) = a b^(x-p) + q = a e^(c(x-p)) + q, c = ln(b). For fitting, Y1 = y1-q, Y2 = y2-q; c = [ln|Y2| - ln|Y1|]/(x2-x1); a = Y1/e^(c(x1-p)).
Worked example
For f(x) = 3 times 2^x, x = 4 gives 48. Setting the function to 24 gives x = 3. The points (0,3) and (2,12) recover a = 3 and b = 2.
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Frequently asked questions
Can the base be one?
Evaluation permits a constant function with base one. Solving for a unique x then fails because the output does not depend on x.
Can fitted values cross zero?
After subtracting the vertical shift q, a non-zero scaled real exponential keeps one sign. Adjusted values with opposite signs or zeros do not fit.
What does the fitting result mean?
The main result is the fitted scale. The page also gives the base, continuous coefficient and equation; it does not establish that future data will follow the curve.