What is the difference between exponential growth and linear growth?

In linear growth the size increases by a constant amount in each time period. For example if you save a fixed amount of $100 every month. In exponential growth the size is multiplied by the same factor in each time period. For example if it grows by 5% each month. Exponential growth starts slowly but eventually outpaces any linear growth.

What is the difference between a discrete model and a continuous model?

In a discrete model the growth rate is applied once per complete cycle, with the result of multiplying the size by the growth rate plus one for each time period. In a continuous model the constant e is used to apply a consistent growth rate, so that at the same growth rate the result is slightly larger.

How do I model decay?

Enter a negative growth rate. For example, entering -5 means that the quantity is decreasing by 5 percent each period. The calculator will give you the half-life, not the doubling time; this is how long it takes for a quantity to reduce by half.

How do you calculate doubling time or half-life?

Both calculations use logarithms. In periodic growth the doubling time is the result of dividing ln(2) by ln(1+r). In continuous growth it is the result of dividing ln(2) by k. To calculate half-life in decay, a negative growth rate is substituted into the same formula resulting in a positive value.

Exponential Growth Calculator

Enter what you know

Growth model

Pick how the change is applied over time.

Periodic — grows once per period

Continuous — grows every instant

Initial value (x₀)

Growth rate per period (r)

Time (number of periods, t)

Final value (x(t))

Fill in any three of the four values (initial, rate, time, final) and leave the fourth blank. The calculator solves for the one you leave empty.

Final value

Growth factor / period
Net change
Total change: %

Doubling time

This is exponential growth: the quantity multiplies by 1.05 each period, a total change of 62.889463%, and doubles about every 14.2067 periods.

Starting from 100, after 10 periods at 5% per period the quantity reaches 162.8895.

Chart and table

Show the growth curve

Plot how the quantity changes over the whole time span.

Show the value table

List the quantity at evenly spaced times, with its multiple of the start.

The quantity at evenly spaced times
Time	Value	Change from start	Multiple of start
0	100	0	1×
1	105	5	1.05×
2	110.25	10.25	1.1025×
3	115.763	15.763	1.1576×
4	121.551	21.551	1.2155×
5	127.628	27.628	1.2763×
6	134.01	34.01	1.3401×
7	140.71	40.71	1.4071×
8	147.746	47.746	1.4775×
9	155.133	55.133	1.5513×
10	162.89	62.89	1.6289×

An exponential growth calculator is a tool that predicts the results when a certain quantity grows by a constant rate every time period. By inputting an initial value, a growth rate and number of periods, the calculator will output the ending value as well as the growth factor, total change, and doubling time. If a negative growth rate is entered, then the same tool functions as an exponential decay calculator and outputs half-life. Since these four main elements are connected by only one equation, any of them can be found when the others are known.

What is exponential growth?

Exponential growth occurs when the rate of growth is proportional to the quantity present. The larger the current size, the faster it grows, resulting in an upward curve that looks like a steep "J". This is different from linear growth, where the same amount is added each time period.

A simple way to distinguish the two is that linear growth is based on addition while exponential growth is based on multiplication. Saving a fixed amount of $100 per month is an example of linear growth. Receiving a constant return of 5 percent per month is an example of exponential growth, since monthly earnings are calculated on the higher balance from the previous month.

Two models: periodic model and continuous model.

There are two standard ways to represent the same concept. The periodic model applies the growth rate only once per complete period, which makes it suitable for annual, monthly or step changes.

x(t) = x_0\,(1 + r)^{t}

Continuous models are appropriate for processes that exhibit a smooth change, such as radioactive decay or continuous compounding interest, because they take into account the rate of change at each instant.

x(t) = x_0\,e^{k t}

In this case x(0) is the initial value, r or k is the growth rate expressed as a decimal number per time period and t is the elapsed time. In both models, a positive growth rate leads to an increase in size while a negative growth rate leads to a decrease in size.

How to use this calculator:

Select a model and enter three of the four main values leaving the fourth blank. If you input the initial value, growth rate expressed as a percentage, and time, then you will get the ending value. If you leave the growth rate blank, you can calculate the growth rate that connects the starting and ending values. If you leave time blank, you can calculate how long it takes for a particular change to happen. If you leave the initial value blank, you can work backwards from a known result. In case of decay, such as entering a negative growth rate like -5, the calculator will give you the half-life.

Understanding of formula:

Suppose a population of 100 cells grows by 5 percent per hour for 10 hours. In the cyclic model, r = 0.05 and t = 10.

x(10) = 100\,(1 + 0.05)^{10} \approx 162.89

Using the same growth rate of 5 percent, the results from the continuous model are slightly higher because the growth is not happening hourly but rather accumulating in small increments.

x(10) = 100\,e^{0.05 \times 10} \approx 164.87

The table below shows the individual symbols, their meaning and examples of values.

Symbol	Meaning	Example
x(0)	Initial value at time 0	100
r or k	Growth rate per period	5 percent
t	Elapsed time (number of periods)	10
x(t)	Final value after time t	162.89

Doubling time and half-life

Doubling time is the length of time it takes for a growing quantity to double in size. Half-life is the corresponding concept when decaying and gives the length of time it takes for a decreasing quantity to reduce by half. Both can be derived from the same logarithm.

t_{\text{double}} = \frac{\ln 2}{\ln(1 + r)} \quad\text{(periodic)}, \qquad t_{\text{double}} = \frac{\ln 2}{k} \quad\text{(continuous)}

At a constant growth rate of 7 percent, the doubling time is ln(2) divided by .07, or about 9.9 periods. As a rule of thumb, there's the Rule of 70: If you divide the percentage growth rate into 70, you can roughly estimate the doubling time.

Exponential growth and linear growth.

In short periods of time it may seem like the linear growth rate is faster, but exponential growth will always eventually overtake that. This is exactly why compound interest, virus spread and rapid population growth can lead to unexpected results. The curve starts out slowly, then suddenly accelerates. If you look at any of the growth curves on this page, you'll see this acceleration immediately.

Applications of the exponential model:

The same equation can be used to model a variety of real-world situations. Bacteria, animals and human populations grow exponentially when resources are plentiful. Money grows through compound interest. In decay, the half-life is used because radioactive substances and drug concentrations in blood decrease exponentially. Moore's Law, the initial spread of epidemics, and the reach of viral posts all follow a similar pattern for some period of time.

Tips for reasonable estimates:

Use the same units for growth rate and time. Match the monthly growth rate to the number of months. Keep unnecessary decimals until the last step to ensure that long-term predictions are not skewed by rounding. Remember that real-world quantities can never grow infinitely. Populations reach a maximum, and markets cool off, so exponential forecasts should be considered short-term reference points rather than long-term promises.

This calculator is for educational purposes and general estimates only. As real systems have limitations that cannot be accounted for in a pure exponential model, the results should be considered guidelines rather than guaranteed values.

Frequently asked questions

What is exponential growth?: Exponential growth is a pattern of data that shows greater increases with passing time, meaning the rate of growth is constantly accelerating. It is the opposite of decay and often used to model things such as population growth when there are no limiting factors. In a discrete model, this results in multiplying by the same factor at equal increments of time. In a continuous model, it involves raising e to a power, with the constant being the rate of growth.*e*used to accumulate change steadily and interest-bearing.
What is the difference between exponential growth and linear growth?: In linear growth the size increases by a constant amount in each time period. For example if you save a fixed amount of $100 every month. In exponential growth the size is multiplied by the same factor in each time period. For example if it grows by 5% each month. Exponential growth starts slowly but eventually outpaces any linear growth.
What is the difference between a discrete model and a continuous model?: In a discrete model the growth rate is applied once per complete cycle, with the result of multiplying the size by the growth rate plus one for each time period. In a continuous model the constant e is used to apply a consistent growth rate, so that at the same growth rate the result is slightly larger.
How do I model decay?: Enter a negative growth rate. For example, entering -5 means that the quantity is decreasing by 5 percent each period. The calculator will give you the half-life, not the doubling time; this is how long it takes for a quantity to reduce by half.
How do you calculate doubling time or half-life?: Both calculations use logarithms. In periodic growth the doubling time is the result of dividing ln(2) by ln(1+r). In continuous growth it is the result of dividing ln(2) by k. To calculate half-life in decay, a negative growth rate is substituted into the same formula resulting in a positive value.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

LibreTexts Mathematics: Exponential Growth and Decay
Open textbook treatment of the exponential growth and decay models.
Nature Scitable: How Populations Grow — The Exponential and Logistic Equations
Where exponential growth applies to populations, and where the logistic limit takes over.