Exponential growth calculator
Enter a start value and rate, see where it ends up.
What is the exponential growth formula?
P = P₀ × (1 + r)^t, where r is the growth rate per period and t the number of periods. 100 growing 7% a period for 10 periods reaches 196.72.
How long does it take to double?
ln 2 ÷ ln(1 + r): at 7% per period, about 10.24 periods. The rule of 72 gives a quick estimate: 72 ÷ 7 ≈ 10.3.
Can it model decay?
Yes — enter a negative rate. At −10% per period, a value halves in about 6.58 periods.
What about continuous growth?
Continuous growth uses P₀e^(kt). This calculator uses growth compounded once per period, as for yearly or monthly rates.
Where is it used?
Populations, investments, bacteria cultures, and anything that grows by a fixed percentage each period.
How many periods does it take to reach a target value?
Solve P = P₀(1 + r)^t for t with logarithms: t = ln(P ÷ P₀) ÷ ln(1 + r). Doubling time is the case P ÷ P₀ = 2. To grow 100 into 500 at 7% per period, t = ln 5 ÷ ln 1.07 = 23.79 periods; enter 100, 7 and 23.79 to confirm that the result is 500.
How do I turn a yearly rate into a monthly one?
Not by dividing by 12. Growth compounds, so the monthly rate is (1 + r)^(1/12) − 1. For 7% a year that is 0.5654% a month, slightly below 7 ÷ 12 = 0.5833%, and twelve months at 0.5654% compound to 1.07.
How does this relate to the e^(kt) form?
They are the same curve. Because b^t = e^((ln b)t), a growth factor b = 1 + r corresponds to k = ln(1 + r) (OpenStax Algebra and Trigonometry 2e, section 6.7). At 7% per period, k = ln 1.07 = 0.06766, so P₀e^(0.06766t) and P₀(1.07)^t give identical values.
The formula and what it assumes
An exponential function f(x) = ab^x multiplies by the same factor b each time x rises by 1; for a percentage rate r per period, b = 1 + r. It grows when b is above 1 and decays when b is between 0 and 1 (OpenStax Algebra and Trigonometry 2e, section 6.1). The calculator takes the rate in percent, accepts anything above −100%, and allows fractional periods. The doubling time is ln 2 ÷ ln(1 + r), and for a negative rate the halving time is ln 0.5 ÷ ln(1 + r). The model assumes the rate stays constant and nothing is added or removed; real populations and markets eventually leave the curve.
Worked example: a population at 1.2% a year
OpenStax Example 3 models India’s population as 1.25 billion in 2013, growing 1.2% per year. For 2031, t = 18, and 1.25 × 1.012^18 = 1.549 billion, matching the book. At that rate the population doubles every ln 2 ÷ ln 1.012 = 58.1 years. For contrast, the book’s bacteria that double every hour (a rate of 100% per period) go from 10 to 10,240 in 10 hours and to 10,485,760 in 20.
Common mistakes
Typing the rate as a decimal: the field wants percent, so 0.07 means 0.07%, and 100 grows only to 100.70 in 10 periods instead of 196.72. Mixing units, such as a monthly rate with a count of years. Splitting a yearly rate evenly across months, which overstates the monthly growth. And forgetting that a decay rate is negative: enter −10, not 10, for a value that shrinks 10% each period.
Sources: OpenStax, Algebra and Trigonometry 2e, sections 6.1 “Exponential Functions” (growth factor, Example 3) and 6.7 “Exponential and Logarithmic Models” (doubling time, half-life, conversion to e^(kt)). Related tools: compound interest calculator, rule of 72 calculator, half-life calculator and log calculator. How we check our calculators.