Exponential Growth Calculator
Four modes: continuous growth/decay (A=Pe^(rt)), half-life decay table, doubling time (t=ln2/r), and compound growth (A=P(1+r)^t). Enter any three values to find the fourth.
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How It Works
Enter any three of: initial value P, rate r (decimal), time t, final value A. The calculator solves for the fourth. Growth: A = P×e^(rt). Decay: A = P×e^(−rt). Select growth or decay using the toggle. A growth rate table at multiple time points also displays. Example: P=1000, r=0.05, t=20 → A=1000×e^1.0=2718.
A = P × e^(rt) [growth] | A = P × e^(−rt) [decay]P=1000, r=5%, t=20yr → A=2718 | P=100, r=−10%, t=5 → A=60.65Enter initial amount and half-life. The calculator shows: amount remaining at any time t, decay constant λ = ln(2)/t½, and a table of values at multiples of the half-life. At t=1×t½: 50% remains. At t=2×t½: 25%. At t=10×t½: 1/1024 ≈ 0.098% remains. Switch to "find half-life" sub-mode to calculate t½ from two measurement points.
A(t) = A₀ × (½)^(t/t½) | λ = ln(2)/t½C-14 t½=5730yr: after 11460yr (2 half-lives) → 25% remainsEnter the growth rate r. Doubling time = ln(2)/r. Also shows: Rule of 70 approximation (70/r%), exact vs approximate comparison, time to triple (ln(3)/r), and time to 10× (ln(10)/r). Enter doubling time to find rate: r = ln(2)/t_double.
T_double = ln(2)/r ≈ 0.693/r | Rule of 70: T ≈ 70/r%r=7%: exact=9.90yr, Rule 70=10.0yr | r=10%: T_double=6.93yrEnter: principal P, rate r per period, number of periods t. A = P×(1+r)^t. Also shows: effective annual rate for different compounding frequencies (monthly: (1+r/12)^12−1), and a growth table. For P=10000, r=6%, t=30: A=10000×1.06^30=57,435.
A = P × (1+r)^t | monthly: P × (1+r/12)^(12t)P=10000, r=6%, t=30yr → A=57,435 | daily compound: A=57,648Growth & Decay mode accepts any three values to find the fourth. To find rate: enter P, A, t, leave r blank. To find time: enter P, A, r, leave t blank. Formulas: r = ln(A/P)/t. t = ln(A/P)/r. These rearrangements use natural logarithm throughout. Example: How long to double at 5%? t = ln(2)/0.05 = 13.86 years.
r = ln(A/P)/t | t = ln(A/P)/r | P = A/e^(rt)A=2P, r=5%: t=ln(2)/0.05=13.86yr to doubleQuick Reference
Verify these in the calculator above.
Growth
A=Pe^(rt), r=7%, t=10
A = P × 2.014
Doubling
Doubling time at 7%
9.9 years
Rule 70
Rule of 70 at 7%
≈ 10 years
Half-life
After 2 half-lives
25% remains
Half-life
After 10 half-lives
≈0.1% remains
Find rate
r from 1000→2500 in 15yr
≈ 6.1%/yr
Compound
Compound 6%, 30yr, $10k
$57,435
Formula
e^0 = ?
1 (any base^0=1)
Tips & Shortcuts
Use Growth & Decay mode to solve for any unknown — enter any three of P, r, t, A and leave the fourth blank.
The Rule of 70 is a quick mental math shortcut: doubling time ≈ 70÷(rate as %). At 2% growth, doubles in ≈35 years.
For monthly compounding: enter r as the monthly rate (annual rate ÷ 12) and t as the number of months. Or use Compound Growth mode which handles compounding frequency.
Half-Life mode shows a decay table at t½, 2×t½, 3×t½, etc. — useful for radioactive dating or pharmacokinetics planning.
Doubling Time mode also shows tripling time (ln(3)/r) and time to 10× (ln(10)/r). Useful for comparing different growth scenarios.
Common Mistakes
Entering rate as percentage instead of decimal
r=7% should be entered as 0.07. Entering 7 gives growth rate of 700% — wildly wrong. Always convert: divide percent by 100 before entering.
Confusing continuous growth with compound growth
Continuous: A=Pe^(rt) gives slightly more than annual compounding A=P(1+r)^t. For 5% over 20 years: continuous gives A=2.718P, annual gives A=2.653P. Use continuous for bacteria, radioactive decay, and "instant" models; use compound for annual/monthly financial calculations.
Using the wrong formula for half-life
A(t) = A₀×(0.5)^(t/t½) or equivalently A₀×e^(−λt) where λ=ln(2)/t½. A common error: using A₀/t instead of the exponential formula. After 2 half-lives: not 0, but (0.5)²=25% remaining.
Mixing time units (years vs months)
If t½ is in days and t is in years, convert to the same unit before calculating. At a half-life of 30 days and measuring after 1 year (365 days): use 365/30=12.17 half-lives → (0.5)^12.17≈0.022% remaining.
Forgetting that doubling time applies to the whole quantity, not just the initial
After one doubling time, the quantity is 2P. After two doubling times, it is 4P (not 3P). Exponential growth compounds — each period's growth is based on the entire current amount, including previous growth.
Frequently Asked Questions
Continuous growth: A = P × e^(rt). P = initial value, r = growth rate (as decimal), t = time, A = final value. For 7% annual growth starting at $1000: after 10 years A = 1000 × e^(0.07×10) = 1000 × e^0.7 ≈ $2014.
Doubling time = ln(2)/r ≈ 0.693/r. For r=7% (0.07): doubling time = 0.693/0.07 ≈ 9.9 years. The Rule of 70 approximates: doubling time ≈ 70/r% (using r as a percentage). At 7%: ≈70/7=10 years.
Continuous growth: A = Pe^(rt) — compounding at every instant. Compound growth: A = P(1+r)^t — compounding at discrete intervals (annually, monthly). Continuous is the mathematical limit of compound as compounding frequency approaches infinity. For same r, continuous gives slightly more growth.
Exponential decay: A = P × e^(−rt). The quantity decreases toward zero but never reaches it. The decay rate r determines the speed. Half-life = ln(2)/r. Examples: radioactive decay, drug concentration in blood, cooling of objects, population decline.
Continuous compound interest: A = Pe^(rt). For nominal rate r=5% over t=20 years: A = P×e^(1.0) = 2.718P. Effective annual rate for continuous compounding: e^r − 1. At 5% nominal: effective = e^0.05 − 1 = 5.127%. Compared to annual compounding: A = P×1.05^20 = 2.653P.
Doubling time ≈ 70/r (where r is in percent). At 2% growth: doubles in ≈35 years. At 10%: doubles in ≈7 years. Rule of 72 (more accurate for higher rates): doubling ≈ 72/r. At 8%: doubling ≈ 72/8 = 9 years. Exact: ln(2)/0.08 = 8.66 years.
Yes. From A = Pe^(rt): r = ln(A/P)/t. Example: $1000 grows to $2500 in 15 years. r = ln(2.5)/15 = 0.916/15 = 0.061 = 6.1%. Enter P, A (as initial and final), and t in Growth & Decay mode to find r.
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