Exponential Growth Calculator

SIMPLIFIED

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Fill in three boxes, then press =.

What the top and bottom each do

The rate goes in as a percentage — 5 means 5%, and −10 means it shrinks by a tenth each period.

Fill in any three — the fourth is worked out.

Amount at the end A

What it grows to. Leave this empty if that is what you want to find.

Amount at the start P

What you begin with — the population, the money, the amount at the start.

Rate each period r %

How much it grows each period. 5 means 5%. A minus sign means it shrinks instead.

How many periods t

How many times the growth happens. This must be in the same unit as the rate — if the rate is per year, this is in years.

A = P(1 + r)^t. Put in any three and leave the fourth empty. A negative rate is decay, and the same formula covers it. Finding t is the one that needs a logarithm.

Examples — tap one and it solves

How it works out, step by step

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₹1,000 at 5% a year becomes ₹1,628.89 after ten years — not ₹1,500. The extra ₹128.89 is growth on growth, and that gap is the whole point of the topic.

This exponential growth calculator works from A = P(1 + r)ᵗ — and unlike most any calculator for exponential growth, it will find whichever quantity you are missing, not just the final amount. Fill in any three of the four and the fourth is worked out — including how long something takes, which is the one that needs a logarithm.

How to Calculate Exponential Growth

One formula does everything:

A = P(1 + r)ᵗ — where P is what you start with, r is the rate each period as a decimal, t is how many periods, and A is what you end with.

The part worth understanding is (1 + r), the growth factor. At 5% it is 1.05, and every period multiplies by it again. Ten periods means 1.05 multiplied by itself ten times, which is 1.6289 — so ₹1,000 becomes ₹1,628.89.

That is why growth beats a straight percentage. Adding 5% ten times gives 50%. Multiplying by 1.05 ten times gives 62.89%, because each year's growth also grows.

To find the rate instead, rearrange. How to calculate exponential growth rate: r is the t-th root of A/P, minus one. From 1,000 to 2,000 in ten periods gives the tenth root of 2 minus 1, about 7.18% a period.

Exponential Growth and Decay: One Formula, Both Ways

Decay is not a different formula. It is the same one with a negative r.

At −10% a period, r is −0.1, so the factor is 0.9 — and multiplying by 0.9 repeatedly shrinks things instead of growing them. 1,000 becomes 590.49 after five periods.

An exponential growth and decay calculator does not need two modes for this. Put a minus sign in front of the rate and the same arithmetic runs, which is why this one has a single set of boxes — an exponential growth or decay calculator and an exponential growth decay calculator are, in the end, the same tool.

Decay has its own useful number: the half-life, how long until only half is left. At −10% a period it is 6.58 periods — and, like doubling time, it does not depend at all on how much you started with. Only the rate matters.

Why Finding the Time Needs a Logarithm

Three of the four come out with ordinary arithmetic. The fourth does not.

To find t you have to get it down from the exponent, and nothing in ordinary algebra does that. A logarithm does:

t = log(A/P) ÷ log(1 + r)

That is not a trick someone invented for this formula. Bringing an exponent down is what logarithms are for, and exponential growth is the reason they exist at all. If you want to see that step on its own, the log calculator works out any logarithm, and the logarithmic equations calculator solves the same kind of equation for x.

The rule of 72 is the shortcut version: divide 72 by the rate for a rough doubling time. At 8% it says 9 periods against a real 9.01, which is close. At 1% it says 72 against a real 69.66, which is not. The calculator gives the exact figure either way.

How to Use This Exponential Growth Calculator

Using this calculator exponential growth question is a matter of four boxes. Fill in three, leave one empty, and press = — the empty one is what gets worked out, and the answer appears in it in blue.

The rate goes in as a percentage: 5 means 5%, and −10 means it shrinks by a tenth each period. The one thing to watch is that r and t must use the same period. If the rate is per year, the time is in years; 18 months is 1.5. Each box has an information button that says so.

Under the answer, one tap shows all four values together and the working step by step, ending with the numbers put back into the formula so you can see it balance. Doubling time or half-life appears without being asked, because it comes free from the rate. It is free to use online, with no account.

Exponential Growth FAQ

How do you calculate exponential growth?

Use A = P(1 + r) to the power t. Multiply the starting amount by the growth factor raised to the number of periods. 1000 at 5% for 10 years is 1000 x 1.05 to the tenth, which is 1628.89.

What is the formula for exponential growth and decay?

The same one. A = P(1 + r) to the power t covers both — a positive r grows, a negative r decays. Minus 10% a year is r = minus 0.1, and the formula needs no changing.

How do you calculate the exponential growth rate?

Rearrange for r: the rate is the t-th root of A over P, minus one. If something goes from 1000 to 2000 in 10 periods, the rate is the tenth root of 2 minus 1, which is about 7.18%.

How do you find how long growth takes?

This is the one that needs a logarithm. t = log(A/P) divided by log(1+r). A logarithm is the only way to bring an exponent down from the top, which is exactly what it was invented for.

What is the rule of 72?

A quick estimate: divide 72 by the percentage rate for the doubling time. At 8% it says 9 periods, and the real answer is 9.01 — close enough for mental arithmetic, but the calculator gives the exact figure.

Start, rate, time, end. Any three give the fourth — and if the missing one is time, that is where the logarithm earns its keep.