Exponential functions model anything that grows or shrinks by a constant percentage: populations, investments, viral spread, radioactive decay and cooling. Two data points are enough to pin one down.
How to use the Exponential Function Calculator
- Mode
- x₁
- y₁
- x₂
- y₂
- a (initial value)
- b (growth factor)
- Evaluate at x
Results update as you type, and everything is calculated in your browser.
How it works
Choose to find the function from two points (both y values must have the same sign) or to evaluate a known function.
Tips
- Logarithms undo exponents: log_b(x) = y means bʸ = x.
- The argument of a logarithm must be positive and the base positive and not 1.
- Use the change-of-base rule to evaluate any logarithm with ln or log₁₀.
- Exponent rules only combine powers with the same base (or the same exponent).
Common mistakes to avoid
- Writing log(a + b) as log a + log b.
- Multiplying exponents when you should add them (a² · a³ = a⁵).
- Forgetting that a⁰ = 1 and a⁻ⁿ = 1/aⁿ.
Inputs explained
| Field | What to enter |
|---|---|
| Mode | Choose one: Find y = a·bˣ from two points, Evaluate y = a·bˣ |
| x₁ | A number |
| y₁ | A number |
| x₂ | A number |
| y₂ | A number |
| a (initial value) | A number |
| b (growth factor) | A number |
| Evaluate at x | A number |
What each result means
- Continuous rate k = ln b = ln([Growth factor b])
- Change per unit of x = ([Growth factor b] – 1)×100
- y at x = [Initial value a]×[Growth factor b]^[Evaluate at x]
- Doubling time (growth) or half-life (decay) = ln(2)÷abs([Continuous rate k = ln b])
Formula
y = a·bˣ: b = (y₂/y₁)^(1/(x₂ − x₁)), a = y₁ / b^x₁k = ln b, doubling time = ln 2 / kWorked example
Inputs — Mode: Find y = a·bˣ from two points; x₁: 0; y₁: 100; x₂: 5; y₂: 200; Evaluate at x: 10.
Results — Growth factor b = 1.148698355; Initial value a = 100; Continuous rate k = ln b = 0.1386294361; Change per unit of x = 14.869835 %; y at x = 400; Doubling time (growth) or half-life (decay) = 5.
Key terms
- Growth factor
- The multiplier b per unit of x.
Frequently asked questions
What makes a function exponential?
The variable is in the exponent: y = a·bˣ.
When is it growth or decay?
Growth when b > 1, decay when 0 < b < 1.
How are b and k related?
b = eᵏ, so k = ln b.
What is the doubling time of 7% growth?
ln 2 / ln 1.07 ≈ 10.24 periods.
What formula does the Exponential Function Calculator use?
y = a·bˣ: b = (y₂/y₁)^(1/(x₂ − x₁)), a = y₁ / b^x₁.