How this instrument works
An exponential function multiplies rather than adds. Starting from a coefficient a, each increase of one unit in x multiplies the running value by the same base b — that single rule, f(x+1) = b · f(x), is really the whole definition, and everything else follows from the ordinary laws of exponents applied to bˣ = b·b·b·… (x factors, when x is a whole number) and extended to fractional and negative x by the same algebra that makes b^(1/2) = √b and b⁻¹ = 1/b consistent.
That constant-ratio structure is what separates an exponential curve from a straight line or a parabola. A linear function adds the same amount for every step in x; an exponential function multiplies by the same factor instead, so the gaps between successive outputs keep growing, or shrinking, rather than staying level. The counter-intuitive payoff shows up for large x: however big the coefficient a is set, and however slowly b exceeds 1, the exponential eventually overtakes any polynomial — x squared, x to the hundred, anything — because multiplying compounds while adding does not.
Two edge cases are worth knowing before you type numbers in. Set x to zero and the base cancels out entirely, since b to the zero equals 1 for any positive b, leaving f(0) = a exactly — the coefficient is the value the function passes through at the origin, not necessarily a 'starting point' in time. Push x negative and the function flips to division, f(-x) = a / bˣ; for a base above 1 the curve then creeps toward zero without ever touching it, a horizontal asymptote rather than an endpoint. The base itself has to stay positive — a negative base raised to a fractional power, like the square root of -4, has no real value, which is why this sheet rejects a base at or below zero.
- Type the starting value into the Coefficient, a field — this is f(0), the output when x is zero.
- Set the multiplier per step in the Base, b field; anything above 1 grows, anything between 0 and 1 decays.
- Enter the exponent in the x field — try a negative number to see the function fall toward zero instead of rising.
- Read the outcome in f(x) = a·bˣ, which recalculates the moment any of the three inputs changes.
Worked example — starting at 100, doubling three times
A lab culture is seeded at a coefficient of 100 cells and, under ideal conditions, doubles every hour, so a base of 2 describes one hour's growth factor. Set x to 3 for three hours and the sheet returns f(3) = 100 × 2³ = 100 × 8 = 800 cells — the population has grown eight-fold, not simply added-to, across three steps.
Nothing about the arithmetic changes if those three hours are instead three rounds of a chain letter, three compounding periods, or three doublings of a shared post — the same f(x) = a·bˣ shape governs all of them, with a as the starting count and b as the doubling factor. Trace the steps by hand: 100, then 100 times 2 equals 200 at hour one, 400 at hour two, and 800 at hour three, the same figure this sheet returns instantly.
Questions
What does the base b control in f(x) = a·bˣ?
The base is the multiplier applied for every unit increase in x. A base above 1 makes the function grow — b=2 doubles each step — a base between 0 and 1 makes it decay, and b=1 freezes the output flat: with a=50 and any x, f(x) = 50 × 1ˣ stays exactly 50, since multiplying by one never changes a value.
How is this different from a linear function?
A linear function adds a fixed amount per step; an exponential function multiplies by a fixed factor per step. Compare the two after three steps from 100: adding 100 each time reaches 400, but multiplying by 2 each time reaches 800, the golden example on this page. The gap between the two only widens as x grows further.
Why does the calculator reject a base of zero or a negative number?
A negative base raised to a fractional exponent has no real value — the square root of -4 would be required for an exponent of one half, and that does not exist among real numbers. A base of zero also breaks the pattern once x drops to zero or below, since zero to the zero is undefined and a negative power of zero means dividing by zero. Keeping b strictly positive keeps every output on this page a genuine real number.
What is f(0) always equal to?
f(0) = a, the coefficient, for any positive base b. That is because any positive number raised to the power zero equals 1, so the base term cancels out and only the coefficient remains. It is the one output value that never depends on which base was chosen.
What mistake do people usually make evaluating a·bˣ by hand?
The most common slip is raising the whole product, a times b, to the power x instead of raising only the base. With a=100, base=2, and x=3, that error gives 200³ = 8,000,000 instead of the correct 100 × 2³ = 800 — the exponent applies to b alone, and the coefficient is multiplied in afterward, not before.
How does a·bˣ relate to the natural exponential eˣ?
Any exponential a·bˣ can be rewritten in base e as a times e raised to the power x times the natural log of b, since b equals e raised to the natural log of b by definition of that logarithm. Calculus favors the e form because the derivative of eˣ is simply itself, while differentiating bˣ directly carries an extra constant factor — the same function, two equivalent faces.