What is the exponential growth formula?

For growth by a fixed percentage each period, P = P₀(1 + r)ᵗ, where P₀ is the starting amount, r is the rate as a decimal and t is the number of periods. For continuous growth, P = P₀eᵏᵗ. Use a negative rate for decay.

How do you calculate exponential growth?

Turn the percentage into a decimal, add 1 to get the growth factor, raise it to the number of periods, and multiply by the starting amount. 1,000 growing 5% a year for 10 years is 1,000 × 1.05¹⁰ ≈ 1,628.89.

How do you find the growth rate from two values?

Divide the final amount by the starting amount, take the t-th root, and subtract 1. If something grows from 1,000 to 2,000 in 10 years, the rate is 2^(1/10) − 1 ≈ 0.0718, about 7.18% a year. Choose 'Rate' under Solve for to do this in the calculator.

How do you calculate exponential decay?

Use the same formula with a negative rate. A car worth $20,000 that loses 15% a year is worth 20,000 × 0.85³ = $12,282.50 after 3 years. The time for the amount to halve is the half-life.

What is the doubling time formula?

For growth of r per period, doubling time = ln 2 / ln(1 + r). For continuous growth at rate k, it's ln 2 / k. At 7% per year, ln 2 / ln 1.07 ≈ 10.24 years. The rule of 72, 72 divided by the percentage, gives a quick estimate: 72 ÷ 7 ≈ 10.3.

What is the difference between P₀(1 + r)ᵗ and P₀eᵏᵗ?

The first grows in steps, once per period, like interest paid yearly. The second grows continuously, every instant. At the same nominal rate continuous growth ends slightly higher: 5% for 10 years gives 1,628.89 one way and 1,648.72 the other.

Exponential Growth Calculator: Growth and Decay

t	amount
0	1000
1	1050
2	1102.5
3	1157.625
4	1215.5063
5	1276.2816
6	1340.0956
7	1407.1004
8	1477.4554
9	1551.3282
10	1628.8946

Populations, investments, viral posts and bacteria cultures all grow the same way: by a percentage of what’s already there, so the bigger they get, the faster they grow. Radioactive material, a car’s resale value and the caffeine in your bloodstream shrink the same way in reverse. That’s exponential growth and decay, and the questions people ask about it nearly always come down to one formula with four quantities. This exponential growth calculator solves for whichever one you’re missing and shows the working.

Exponential growth formulas: P equals P sub 0 times (1 plus r) to the t, and P equals P sub 0 times e to the k t — Growth by a percentage per period, and continuous growth

How to use the exponential growth calculator

Choose what to solve for: the final amount, the starting amount, the rate or the time.
Choose the model. is for a fixed percentage per period, like annual interest or a population growing 3% a year. is for continuous growth.
Enter the other three values. Type the rate as a percentage, 5 for 5%, and use a negative number for decay. The time can be in any unit as long as it matches the rate: years with a yearly rate, hours with an hourly rate.
Read the results: the answer, the growth factor per period, the doubling time or half-life, a table of values and a graph with your point marked.

The exponential growth formula

is the starting amount, the growth rate per period as a decimal, the number of periods and the amount after periods. The number is the growth factor: what the amount is multiplied by each period. A 5% increase is a factor of 1.05; a 15% decrease is a factor of 0.85. The exponential growth and decay lesson explains where the formula comes from.

Worked example: 5% a year for 10 years

This is the calculator’s first preset: 1,000 growing 5% a year.

Step 1 — the growth factor. .

Step 2 — apply it 10 times.

The amount grew by 628.89, not by 500, which is what ten separate 5% steps of the original 1,000 would give. Each year’s 5% is taken of a larger amount than the year before, and that compounding is exactly what makes growth exponential.

Solving for the rate or the time

The same formula answers the reverse questions; you just solve it the other way round.

How fast? A culture grows from 50 to 3,200 bacteria in 6 hours. Divide to get the total factor, , then take the 6th root: , so , or 100% an hour. The culture doubles every hour.

How long? How long does 1,000 take to reach 2,000 at 7% a year? Taking logarithms of gives

That’s the doubling time, and it doesn’t depend on the starting amount at all. Solving exponential equations covers the logarithm step in detail.

Doubling time and the rule of 72

Bars comparing exact doubling times with the rule of 72: at 2% the exact time is 35 periods versus 36 by the rule; at 4%, 17.67 versus 18; at 6%, 11.9 versus 12; at 8%, 9.01 versus 9; at 12%, 6.12 versus 6 — The rule of 72 is a good estimate across everyday rates

The exact doubling time is , which the calculator reports on every growth problem. For mental arithmetic, the rule of 72 says divide 72 by the percentage rate. At 8% the rule gives 9 periods and the exact answer is 9.01, almost identical. It drifts a little at very low or very high rates, but it’s close enough to sanity-check any result.

Exponential decay

Decay is growth with a negative rate. A car that loses 15% of its value each year keeps 85% each year:

After 3 years it’s worth . For decay the calculator reports the half-life instead of the doubling time: years for this car to lose half its value. For radioactive decay problems given in terms of half-lives, the half-life calculator works directly from the half-life.

Percent per period vs continuous growth

Model	Formula	1,000 at 5% for 10 periods
once per period		1,628.89
continuous		1,648.72

Continuous growth adds its increase every instant rather than once per period, so it ends slightly higher at the same nominal rate. Its effective rate per period is , which the calculator shows: a continuous 5% is equivalent to about 5.13% per period. Bank interest that compounds continuously, population models and many science formulas use the form; see continuous compound interest for the money version.

Reading the graph

The curve in the calculator always passes through the starting amount at and bends more steeply as time goes on for growth, or flattens toward zero for decay. It never reaches zero, however long a decay runs. The dashed marker shows where the amount has doubled or halved, and the amber point is your answer. If your point sits somewhere surprising, such as below the starting amount on a growth problem, check the sign of the rate first.

Where exponential growth shows up

Savings and loans: a balance growing at a fixed annual rate.
Populations: cities, animals or bacteria growing by a steady percentage, at least until resources run short, which is when logistic growth takes over.
Depreciation: cars and equipment losing a fixed share of their value each year.
Medicine: a drug clearing from the body at a fixed rate per hour.
Sequences: the values at whole-number times form a geometric sequence with ratio , which the geometric sequence calculator handles term by term.

Common mistakes

Using the percentage instead of the decimal. 5% is , so the factor is 1.05, not 6. (In this calculator, type 5 and it converts for you.)
Using instead of . The factor for 5% growth is 1.05; is nearly zero.
Treating a decrease as with a positive . A 15% loss is a factor of 0.85.
Mismatched units. A yearly rate needs time in years. Convert a monthly rate before mixing it with years.
Adding instead of multiplying. Ten years at 5% is not 50% growth; it’s 62.9%.

Exponential Growth Calculator