Choose which variable to solve for, enter the other three, and watch the result and the fully substituted formula update as you type.
| Symbol | Formula | Solves For |
|---|---|---|
| x(t) | x(t) = x0(1 + r)t | Final value from x0, r, and t |
| x0 | x0 = x(t) / (1 + r)t | Initial value from x(t), r, and t |
| r | r = (x(t) / x0)1/t − 1 | Growth rate from x(t), x0, and t |
| t | t = ln(x(t) / x0) / ln(1 + r) | Time from x(t), x0, and r |
What the table shows: all four rows describe the same relationship, so they must round-trip back to the same numbers. At the calculator's default inputs, x0 = 1,000, r = 6%, t = 12, solving for x(t) gives 2,012.20; feeding that 2,012.20 back in and solving for x0 returns 1,000 again, solving for r returns 6.00%, and solving for t returns 12.00. Four formulas, one fact.
The exponential growth formula is x(t) = x0(1 + r)t, tying together four variables, final value, initial value, growth rate, and elapsed time, such that any three of them algebraically determine the fourth.
x(t) is the value being solved for at some point in the future, x0 is the value the process started from, r is the fixed proportional rate applied every period, and t counts how many periods have passed. Nothing else appears in the equation. Every quantity a growth problem could ask about reduces to one of these four symbols.
Because it is a single equation, the formula behaves like a small machine with four slots: fill in any three and the fourth is no longer free to be anything. It is fixed by the other three. This page exists as the lookup for that machine, cataloging all four ways of rearranging it side by side rather than leaving anyone to re-derive one from scratch each time a different variable is the unknown.
Most people meet this formula in its forward direction first, the one used by the calculator on this site's homepage: start from a known x0 and r, project forward through t periods, and read off x(t). The three other rearrangements below run the same relationship in reverse, which is exactly what's needed whenever the known quantity is the outcome rather than the starting conditions.
The exponential growth formula has exactly four rearrangements, one for each variable it contains, and each is a direct algebraic consequence of the same starting equation. Two are reached with plain multiplication and division; the other two need a logarithm.
The original form. Use it whenever the starting value, the rate, and the elapsed time are all already known.
Use it to back out a starting value from a known outcome, its rate, and the time it took to get there.
| Symbol | Formula | Isolates | Needs a Logarithm? |
|---|---|---|---|
| x(t) | x(t) = x0(1 + r)t | Final value | No |
| x0 | x0 = x(t) / (1 + r)t | Initial value | No |
| r | r = (x(t) / x0)1/t − 1 | Growth rate | No, a fractional exponent instead |
| t | t = ln(x(t) / x0) / ln(1 + r) | Time | Yes |
What the table shows: x(t) and x0 sit outside the exponent in the original equation, so isolating either one only ever takes multiplying or dividing by (1 + r)t. r and t sit inside the exponent, so pulling either one free takes an extra step, a fractional power for r, a true logarithm for t. The next section derives both of those extra steps in full.
Every rearrangement of the growth formula comes from the same three algebraic moves applied to x(t) = x₀(1 + r)t: dividing both sides, raising both sides to a power, and taking a logarithm of both sides.
x(t) = x₀(1 + r)t is the single equation every rearrangement below is derived from.
x(t) / (1 + r)t = x₀, which is the x₀ rearrangement written the usual way round.
x(t) / x₀ = (1 + r)t. The growth factor and elapsed time are now alone on the right.
(x(t) / x₀)1/t = 1 + r, since raising a power to 1/t cancels the outer exponent t. Subtracting 1 gives r = (x(t) / x₀)1/t − 1.
Starting again from x(t) / x₀ = (1 + r)t, the log power rule turns the right side into t·ln(1 + r), giving ln(x(t) / x₀) = t·ln(1 + r). Dividing by ln(1 + r) gives t = ln(x(t) / x₀) / ln(1 + r).
A vineyard harvested 1,200 crates in its first season and 1,890 crates eight seasons later, with no change in acreage. The r-rearrangement finds the steady per-season growth rate that accounts for the increase.
Checking the answer by running it back through the original formula confirms it: 1,200 × (1 + 0.058425)8 = 1,890 crates, matching the actual eight-season result.
Which variable to solve for follows directly from which one is missing: solve for x(t) to project forward, x₀ to project backward, r to measure a rate that already happened, and t to measure how long a change took. All four questions use the identical equation, only the arrangement changes.
Solving for x(t) answers "where will this end up?", a savings balance after a fixed number of years, a population after a fixed number of generations, given a known starting point and rate. Solving for x₀ answers the opposite question, "where did this start?", useful whenever only the current value, the rate, and the elapsed time are on hand, and the original figure has to be reconstructed rather than measured directly.
Solving for r answers "what rate got me here?", the question behind measuring an actual historical growth rate from two observed values a known number of periods apart, rather than assuming a rate up front. Solving for t answers "how long until, or how long did it take?", doubling time, payback periods, or how many cycles separate two known measurements.
Because r and t both require a logarithm to isolate, working either of them out by hand benefits from seeing the full step-by-step process rather than just the closed-form result above, that walkthrough lives on the exponential equation solver, alongside a rate-only version tuned specifically for the "what rate got me here" question on the rate calculator. For the forward direction, x₀, r, and t known, x(t) unknown, the exponential growth calculator on the homepage remains the fastest route to an answer.
x(t) is the value after t periods have passed, x₀ is the starting value at time zero, r is the growth rate per period expressed as a decimal, and t is the number of periods that have elapsed. Every rearrangement on this page uses the exact same four symbols, only which one sits alone on the left side changes.
A rate entered as a percentage, such as 6%, is always converted to r = 0.06 before it goes into any of the formulas.
The growth formula is one equation containing four variables, and a single equation can only be solved for one unknown once every other term inside it has a specific numeric value. With two or more unknowns still in the equation there are infinitely many combinations that satisfy it, so no unique answer exists.
This is why each rearrangement on this page is written as "solve for X given the other three" rather than "solve for X given some of the others."
Because r and t both sit inside the exponent of (1 + r)t, and a logarithm is the only algebraic operation that can move a variable out of an exponent. x(t) and x₀ sit outside the exponent, so isolating either one needs nothing more than multiplication or division.
The exponential equation solver walks through that logarithm step in full, one line at a time, for anyone solving for r or t by hand.
The homepage calculator only computes forward, from x₀, r, and t to x(t); this page's calculator solves in whichever direction you choose, including backward from x(t) to any of the other three. Both tools use the identical underlying formula. This page simply exposes all four of its rearrangements instead of just one.
Anyone who already has a final value and needs to work backward to a rate or a starting point is in the right place here rather than on the homepage.
Yes. The continuous form x(t) = x₀ekt has the same four rearrangements, with a natural logarithm replacing the (1 + r)-based logarithm used here. Solving it for k gives k = ln(x(t)/x₀) / t, and solving it for t gives t = ln(x(t)/x₀) / k, mirroring the discrete forms almost symbol for symbol.
The continuous growth reference covers that version and its constant e in full.
Solving for r or t breaks down if x₀ or x(t) is zero or negative, because both of those rearrangements divide one by the other and then take a logarithm, and logarithms are undefined for zero or negative inputs. Solving for x(t) or x₀ has no such restriction. Those two rearrangements only multiply and divide, so they accept a negative rate or a zero starting value without error.
The calculator above reflects this directly: pick "r" or "t" with an invalid pair of values and the readout shows a dash instead of a number.
Solving for x(t) projects a future value from a known rate; solving for x₀ backs out a starting point from a known result; solving for r measures the rate a change actually happened at; and solving for t answers how long a given change took. A retailer projecting next year's revenue solves for x(t); an appraiser backing out a property's purchase price from its current value solves for x₀.
The rate solver is built specifically around that third case, with more worked examples than fit on this reference page.