Math
Exponential Function Calculator
Work with any exponential function y = a·b^x.
Evaluate it at a value of x with a table of nearby values, find a and b from two points on the curve, or solve for the x that produces a given y using logarithms. Each answer says whether the function grows or decays and by what percent per step.
b > 0 and b ≠ 1. Above 1 grows; between 0 and 1 decays.
Try an example
Result
Result
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- Function
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- Type
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- Percent change per unit x
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- Table of values
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
Anatomy of y = a·b^x
| Part | Role | How to read it |
|---|---|---|
| a | starting value | y-intercept; the curve passes through (0, a) |
| b | base / growth factor | b > 1 grows, 0 < b < 1 decays |
| b - 1 | rate per unit | b = 1.07 means +7% each step; b = 0.9 means -10% |
| x | input | each +1 in x multiplies y by b |
Finding the function from two points
Divide the two equations y₂ = a·b^x₂ and y₁ = a·b^x₁ to cancel a: y₂/y₁ = b^(x₂ - x₁). Take the (x₂ - x₁)-th root to get b, then substitute back into either point to get a. From (1, 6) and (3, 24): b = (24/6)^(1/2) = 2 and a = 6 / 2 = 3, so y = 3·2^x.
Solving a·b^x = y for x
Growth vs decay
- y = 3·2^x doubles each step: growth.
- y = 100·(0.5)^x halves each step: decay.
- y = 5·(1.05)^x grows 5% per step, the same as a 5% interest rate.
- Any decay y = a·b^x can be written as growth with a negative x direction, since b^x = (1/b)^(-x).
How to use it
- Evaluate: enter a, b, and x to get y with the surrounding table of values.
- Two points: enter (x₁, y₁) and (x₂, y₂) to recover a and b. The primary result is b.
- Solve for x: enter a, b, and the target y; the calculator takes a The exponent you would need to raise a base to in order to get a number..
How to read the answer
a is the y-A point where a graph crosses an axis. (the value at x = 0). b is the factor y is multiplied by each time x increases by 1; b - 1 as a percent is the growth or decay rate. If b > 1 the function grows; if 0 < b < 1 it decays toward 0.
Common mistakes and edge cases
- Confusing a·b^x with (ab)^x. Only b is raised to the power.
- Using a base of 1 or a negative base; the function needs b > 0 and b ≠ 1.
- Expecting the curve to cross the x-axis; an exponential never reaches 0.
- Solving for x by dividing by b instead of taking a logarithm.
Worked examples
Evaluate
y = 3·2^x at x = 4
Result
48
Decay
y = 100·(0.5)^x at x = 3
Result
12.5
Two points
Through (1, 6) and (3, 24): b = 2
Result
2
Two points, decay
Through (0, 80) and (2, 20): b = 0.5
Result
0.5
Solve for x
3·2^x = 96 gives x = 5
Result
5
Solve for x, decimal
5·1.05^x = 10 (doubling at 5%)
Result
14.206699
Negative x
y = 3·2^x at x = -2
Result
0.75
Impossible
Base 1 is not exponential
Result
Error
Frequently asked questions
What is an exponential function?+
A function of the form y = a·b^x where the variable is in the exponent. Each unit increase in x multiplies y by the same factor b.
How do you find an exponential function from two points?+
Divide the y-values to cancel a: y₂/y₁ = b^(x₂ - x₁). Take the root to get b, then a = y₁ / b^x₁. Use the two-points mode for the full work.
How do I know if it is growth or decay?+
Look at the base: b > 1 is growth, 0 < b < 1 is decay. The percent rate is (b - 1) × 100%.
How do you solve 3·2^x = 96?+
Divide by 3 to get 2^x = 32, then x = log₂(32) = 5. For non-exact cases, x = ln(y/a) / ln(b).
Why can't the base be negative or 1?+
A negative base is undefined for most fractional x, and base 1 gives a flat line y = a. Both break the exponential shape.
What is the y-intercept of y = a·b^x?+
a, because b^0 = 1. The curve always passes through (0, a).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026