Exponential Function Calculator

Background.

Some quantities grow by adding the same amount each step; others grow by multiplying by the same factor. The second kind is exponential, and it behaves very differently. Bacteria doubling every hour, money compounding at 5% a year, the share of a radioactive sample left after each half-life: all follow f(x) = a · bˣ, where a is the starting amount and b the factor applied at every step.

The gap between adding and multiplying starts small and becomes enormous. Start at 3 and add 3 each step, and after 10 steps you have 33. Start at 3 and double each step, and after 10 steps you have 3,072. The early terms look alike; it is the repeated multiplication that runs away.

Whether the function grows or shrinks depends only on b. A factor above 1 means growth, and b − 1 is the rate per step, so b = 1.05 is 5% growth. A factor between 0 and 1 means decay: b = 0.5 halves the value each step. b = 1 leaves it unchanged. The factor must be positive, because a negative base raised to a fractional power has no real value.

The exponent need not be a whole number, or even positive. A fractional x lands between the steps — 3 · 2^0.5 is about 4.243 — and a negative x runs the process backwards, so 3 · 2⁻² = 0.75 is where the quantity stood two doublings before the start. If your model uses a continuous rate, written a · e^(kx), enter b = e^k: a continuous 5% rate is a factor of about 1.0513.

What is exponential function calculator?

An exponential function of the form f(x) = a · bˣ has a constant ratio between successive values: every time x rises by 1, the output is multiplied by b. The constant a is the value at x = 0, the graph's y-intercept, and it sets the scale and the sign of every result. The variable sits in the exponent, which is what separates it from a power function such as x², where the base varies and the exponent is fixed.

How to use this calculator.

1. 01Enter Initial value a, the amount at x = 0 — a starting balance, population or dose. It may be zero or negative; a negative a mirrors the curve below the horizontal axis.
2. 02Enter Growth factor b, the multiplier per step: 1 plus the rate for growth (1.08 for 8%), 1 minus the rate for decay (0.85 for a 15% loss each step). It must be greater than zero.
3. 03Enter Exponent x, the number of steps from the starting point, in the unit the factor refers to — years if b is a yearly factor. Fractions and negative values from −100 to 100 are accepted.
4. 04Read f(x). To work backwards and find how many steps a target value takes, use logarithms: x = log(f ÷ a) ÷ log(b).

The formula.

f(x) = a × b^x. The power is applied first: b is raised to x, and the result is then scaled by a. For a whole positive x, b^x is b multiplied by itself x times; for a negative x it is 1 ÷ b^|x|; for a fractional x it is a root, so b^0.5 is √b. Because each unit of x multiplies the output by b, the growth factor dominates the answer: moving b from 2 to 3 at x = 10 turns 2^10 = 1,024 into 3^10 = 59,049. The initial value only stretches the curve and sets its sign. Results are shown to at most 10 decimal places, so a value that has decayed far enough, such as 3 × 0.5^100, displays as 0.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.