Exponential Growth Calculator

Calculate exponential growth and decay using continuous (ert) or discrete ((1+r)t) models. Find doubling time, growth factor, and timeline projections.

Select Growth Model & Enter Values

A(t) = A0 · ert

Result

Final Value A(t)
---
Initial Value (A0) ---
Growth Rate ---
Time ---
Growth Factor ---
Total Change ---
Percent Change ---
Doubling Time ---

Step-by-Step Solution

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Understanding Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value. This leads to accelerating growth over time -- the larger the quantity becomes, the faster it grows. It is one of the most important mathematical concepts with applications in biology, finance, physics, and many other fields.

Growth Models

Continuous Growth

Uses the natural exponential function with Euler's number e for continuously compounding growth.

A(t) = A0 * e^(rt)

Discrete Growth

Growth that occurs in regular intervals or periods, like annual or monthly.

A(t) = A0 * (1 + r)^t

Doubling Time (Continuous)

The time it takes for a quantity to double under continuous growth.

td = ln(2) / r

Doubling Time (Discrete)

The number of periods it takes for a quantity to double under discrete growth.

td = ln(2) / ln(1 + r)

Rule of 70

A quick approximation: doubling time is approximately 70 divided by the percentage growth rate.

td approx 70 / (r * 100)

Exponential Decay

When the rate is negative, the quantity decreases over time. The half-life is the time to reach 50%.

Half-life = ln(2) / |r|

Real-World Applications

  • Population Growth: Modeling how populations grow when resources are unlimited.
  • Compound Interest: How savings and investments grow over time with reinvested earnings.
  • Radioactive Decay: The rate at which unstable atoms undergo radioactive disintegration.
  • Bacterial Growth: Under ideal conditions, bacterial colonies can double at regular intervals.
  • Drug Metabolism: How medications are eliminated from the body over time.
  • Inflation: The decrease in purchasing power of money over time.

Growth vs. Decay

When the rate r is positive, the model describes growth (the quantity increases over time). When r is negative, it describes decay (the quantity decreases). For the discrete model, growth occurs when r > 0 (base > 1) and decay when -1 < r < 0 (0 < base < 1).

Tips for Using This Calculator

  • Enter the growth rate as a decimal: 5% = 0.05, not 5.
  • For decay, use a negative rate (e.g., -0.03 for 3% decay).
  • Continuous growth is slightly faster than discrete growth at the same nominal rate.
  • The doubling time is independent of the initial value -- it only depends on the rate.
  • Use the Rule of 70 for quick mental estimates of doubling time.