Enter your initial value (x₀), growth rate (r%), and time period (t) to model exponential growth or decay. The Exponential Growth Calculator applies the formula x(t) = x₀ × (1 + r/100)ᵗ and returns the final value, total growth, and a growth schedule table showing how the quantity evolves at each step.
Final Value x(t)
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Total Growth / Decay
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Growth Factor
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Total % Change
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The exponential growth formula is x(t) = x₀ × (1 + r/100)ᵗ, where x₀ is the initial value, r is the growth rate in percent, and t is the number of time periods. When r is positive the quantity grows; when r is negative it decays. The formula models any process that compounds at a constant percentage rate.
Yes — a negative time value means you are looking backward from the reference point t = 0. Plugging a negative t into the formula gives you the value the quantity had before the starting point, assuming the same constant rate applied in reverse. This is useful in radiocarbon dating and similar back-calculation scenarios.
When t = 0, any base raised to the power of zero equals 1, so x(0) = x₀ × 1 = x₀. In other words, the final value is exactly the initial value — no growth or decay has occurred yet. This confirms that x₀ is always the starting quantity at time zero.
Linear growth adds a fixed amount each period (e.g. +10 every year), while exponential growth multiplies by a fixed factor each period (e.g. ×1.1 every year). The key difference becomes obvious over long timeframes: exponential growth accelerates continuously, whereas linear growth increases at a steady, constant pace. Most natural processes — population, compound interest, viral spread — follow exponential rather than linear patterns.
Enter a negative growth rate — for example, -5 for a 5% decay rate per period. The formula still works identically: x(t) = x₀ × (1 − 0.05)ᵗ. The value will shrink each period instead of growing. Note that the decay rate must stay between -100% and 0%; a rate below -100% would produce a negative quantity, which is physically meaningless for most applications.
Exponential growth and decay appear across many fields: population biology (bacteria doubling), finance (compound interest), physics (radioactive decay), epidemiology (disease spread), and technology (Moore's Law). Any process where the rate of change is proportional to the current value follows an exponential pattern. This calculator can model all of these by adjusting the initial value, rate, and time unit.
Even small differences in growth rate produce dramatically different outcomes over long periods. For example, starting with 100 units, a 5% rate for 50 periods yields about 1,147, while a 10% rate yields about 11,739 — more than ten times larger. This compounding effect is why early investment returns and early interventions in population control have outsized long-term impacts.
Rearrange the formula to solve for t: t = ln(x_target / x₀) / ln(1 + r/100). For example, to double your initial value (x_target = 2 × x₀) at a 7% rate, t = ln(2) / ln(1.07) ≈ 10.24 periods. This is the mathematical basis of the Rule of 72, which estimates doubling time as approximately 72 divided by the annual percentage rate.