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Exponential Growth Calculator

Calculate exponential growth and decay: P(t) = P₀ × e^(rt). Model population growth, compound interest, radioactive decay, and more.

By Christopher FloiedPublished Updated

Positive for growth, negative for decay

Results

Final Value P(t)

1648.7213

Doubling Time

13.8629

Half-Life

N/A (r > 0; growing)

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Exponential Growth Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.

2

Review your inputs

Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Exponential Growth Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Exponential Growth Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Exponential Growth Calculator

Exponential growth and decay describe processes where the rate of change is proportional to the current value. The formula P(t) = P₀e^(rt) models continuous exponential change, where P₀ is the initial value, r is the growth rate (positive for growth, negative for decay), and t is time. This model applies to an extraordinary range of phenomena: bacterial growth, compound interest, radioactive decay, population dynamics, viral spread, drug metabolism, and Newton's law of cooling. The doubling time (how long until a growing quantity doubles) and half-life (how long until a decaying quantity halves) are constant for exponential processes — a distinctive property. This calculator computes the final value along with the doubling time or half-life.

The Math Behind It

The exponential growth model derives from the differential equation dP/dt = rP, whose solution is P(t) = P₀e^(rt). The doubling time T₂ = ln(2)/r ≈ 0.693/r. The Rule of 70 approximates this: T₂ ≈ 70/percentage_rate. For decay (r < 0), the half-life T½ = ln(2)/|r|. Discrete compound growth uses P(t) = P₀(1 + r/n)^(nt), which approaches P₀e^(rt) as n → ∞ (continuous compounding). Exponential growth cannot continue indefinitely in physical systems; logistic growth P(t) = K/(1 + ((K−P₀)/P₀)e^(−rt)) adds a carrying capacity K. The exponential model is the first-order Taylor approximation of any growth process near its initial state.

Formula Reference

Exponential Growth

P(t) = P₀ × e^(rt)

Variables: P₀ = initial value, r = rate, t = time

Doubling Time

T₂ = ln(2) / r

Variables: r = growth rate (r > 0)

Half-Life

T½ = ln(2) / |r|

Variables: r = decay rate (r < 0)

Worked Examples

Example 1: Population Growth

A population of 1000 grows at 5% continuously. Find the population after 10 years.

Step 1:P(10) = 1000 × e^(0.05 × 10)
Step 2:= 1000 × e^0.5
Step 3:= 1000 × 1.6487

P(10) ≈ 1648.7

Common Mistakes & Tips

  • !Confusing continuous rate r with discrete rate — 5% continuous growth is not the same as 5% annual compounding.
  • !Using a positive rate for decay problems (should be negative).
  • !Forgetting that the doubling time formula only applies when r > 0.

Related Concepts

Exponential Function

The general exponential function f(x) = a × bˣ.

Natural Log

ln is the inverse of the exponential function.

e Calculator

Compute e^x directly.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

e Calculator (e^x)

Exponential Function Calculator

Geometric Sequence Calculator

Natural Log Calculator

Frequently Asked Questions

What is the Rule of 70?

The Rule of 70 estimates doubling time: divide 70 by the percentage growth rate. At 7% growth, the doubling time is approximately 70/7 = 10 periods.

How does continuous growth differ from compound growth?

Continuous growth uses P₀e^(rt); compound growth uses P₀(1+r/n)^(nt) where n is compounding frequency. As n → ∞, compound growth approaches continuous growth.

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