# Exponential Function Calculator

URL: https://speedycalculator.com/calculators/math/exponential-function/
Last updated: 2026-10-30

> The exponential function calculator finds the equation y = a·bˣ (and y = a·eᵏˣ) through two points, or evaluates a given exponential function at any x, with growth or decay rate and doubling or half-life.

## Inputs

- Mode
- x₁
- y₁
- x₂
- y₂
- a (initial value)
- b (growth factor)
- Evaluate at x

## Results

- Growth factor b
- Initial value a
- Continuous rate k = ln b
- Change per unit of x
- y at x
- Doubling time (growth) or half-life (decay)

## Formula

```
y = a·bˣ:  b = (y₂/y₁)^(1/(x₂ − x₁)),  a = y₁ / b^x₁
k = ln b,  doubling time = ln 2 / k
```

## Explanation

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

FieldWhat to enterModeChoose one: Find y = a·bˣ from two points, Evaluate y = a·bˣx₁A numbery₁A numberx₂A numbery₂A numbera (initial value)A numberb (growth factor)A numberEvaluate at xA 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])

## Worked 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.

## FAQ

**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.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – Exponential function](https://mathworld.wolfram.com/ExponentialFunction.html)