Choose Evaluate to compute bx directly, or Solve for x to invert the relationship and recover the exponent from a known result, updating instantly as you type.
| x | bx |
|---|---|
| 0 | 1.00 |
| 1 | 1.50 |
| 2 | 2.25 |
| 3 | 3.38 |
| 4 | 5.06 |
| 5 | 7.59 |
| 6 | 11.39 |
| 7 | 17.09 |
| 8 | 25.63 |
What the table shows: at the default base of 1.5, the curve is nearly flat near x = 0, rising from 1.00 to just 2.25 between x = 0 and x = 2, then climbs increasingly steeply, more than tripling from 7.59 at x = 5 to 25.63 at x = 8. If the base were instead a fraction between 0 and 1, such as 0.6, the same table would fall toward zero rather than rise, flattening out just above the horizontal axis instead of shooting upward.
An exponential function is f(x) = bx, a rule where the input x is the exponent rather than the base, the defining feature that separates it from a polynomial, where the exponent is fixed and the base is what varies.
Because x sits in the exponent, increasing x by one full unit does not add a fixed amount to the output. It multiplies the output by a fixed factor, b. That single difference in mechanics is responsible for the entire shape of the curve: additive change produces a straight line, but multiplicative change produces a curve that bends away from the horizontal axis and accelerates the further it travels.
The base b decides which way the curve bends. When b is greater than 1, each step multiplies the previous output by more than itself, so the function grows, slowly at first, then rapidly. When b sits between 0 and 1, each step multiplies by a factor smaller than 1, so the function decays toward zero instead. A base of exactly 1 produces neither: 1x equals 1 for every x, a flat line with no rate of change at all.
Every exponential curve passes through the same fixed point regardless of its base: f(0) = b0 = 1. From that shared starting point at (0, 1), the base alone determines whether the curve climbs to the right or falls toward zero, and how sharply it does either.
The domain of f(x) = bx is every real number, its range is every value greater than 0 for a positive base, and its graph has a horizontal asymptote at y = 0 that it approaches but never touches. These three facts together describe the entire shape of the curve without needing to plot a single point.
There is no real number x for which bx is undefined, so long as the base b is positive. Negative, fractional, and enormous exponents all produce a valid result.
No exponent, applied to a positive base, can ever produce zero or a negative number. The output can get extremely small or extremely large, but it never crosses the x-axis.
| x | y = 2x |
|---|---|
| -1 | 0.5000 |
| -2 | 0.2500 |
| -3 | 0.1250 |
| -5 | 0.0313 |
| -10 | 0.0010 |
What the table shows: as x moves from −1 to −10, y keeps shrinking, from 0.5000 down to 0.0010, but it never reaches exactly zero, no matter how negative x becomes. That flat line the curve keeps approaching without ever touching is the horizontal asymptote, and it sits at y = 0 for every exponential function with a positive base.
To evaluate bx, raise the base to the given power directly; to invert the relationship and solve bx = y for x, take the natural log of both sides and divide by ln(b).
Identify b, the fixed multiplier, and x, the power it is being raised to.
Multiply b by itself x times for a whole-number exponent, or use a calculator's power function for fractional or negative exponents, to get the result y.
When y = bx is known but x is not, the goal shifts from computing a result to solving for the exponent that produced it.
ln(y) = ln(bx) = x·ln(b), using the logarithm power rule to bring the exponent down as a multiplier.
x = ln(y) / ln(b) isolates the exponent and gives its exact value.
An audio engineer chains 7 identical amplifier stages, each multiplying the incoming signal's amplitude by a fixed factor of 2.5. The exponential function gives the total gain after all 7 stages. The same relationship, run in reverse, recovers the stage count from a measured gain.
Feeding the computed gain of 610.35 back through the inverse formula returns exactly 7, confirming that evaluating and inverting are two directions of the same relationship, not two different calculations.
The base b and a periodic growth rate r describe the same multiplier through b = 1 + r, so a base of 1.05 is the same relationship as a 5% growth rate. Multiplying by 1.05 every step and growing by 5% every step are not two different processes. They are the same process described in two notations.
This page uses b and x, pure base-and-exponent notation, with no assumptions about what x represents, because f(x) = bx is a general mathematical function, not a model tied to any particular quantity. The rest of this site frames the identical mathematics as a growth model instead, swapping b and x for x₀, r, and t: an initial value growing at a periodic rate over elapsed time. The growth formula reference lays out that notation in full, and the continuous growth calculator covers the version where the multiplier compounds every instant rather than once per period.
The default base on this page, b = 1.5, is exactly a 50% growth rate (r = 0.5). Each step multiplies the previous value by one and a half times itself. A base below 1, such as b = 0.9, converts the same way to a negative rate: r = −0.10, a 10% decline per step. Anyone who has already used the exponential growth calculator on this site's front page has entered r directly; this page's calculator asks for the same underlying number, just expressed as b instead.
A function is exponential when its input variable sits in the exponent rather than in the base, as in f(x) = bx. That single placement is what separates it from a polynomial like f(x) = x3, where the exponent is fixed and the base is what varies.
Because x is the exponent, each unit increase in x multiplies the output by the same factor b, rather than adding the same amount. That difference is the source of the curve's characteristic accelerating shape.
The horizontal asymptote at y = 0 means the curve gets arbitrarily close to zero as x moves toward negative infinity (for b > 1), but never actually reaches or crosses it. No value of x, however large in magnitude, makes bx equal zero or negative when b is positive.
This is why the range of f(x) = bx is written as y > 0 rather than y ≥ 0. Zero is a limit the function approaches, not a value it takes.
The base must be positive because a negative base raised to a fractional exponent, such as (−2)0.5, has no real-number result, which would leave gaps in the domain. Requiring b > 0 keeps bx defined for every real x, with no exceptions.
The base also cannot equal 1, since 1x = 1 for every x, a flat horizontal line, not a growing or shrinking curve, and a case where solving bx = y for x has either no solution or infinitely many.
A base greater than 1 always produces growth, and a base between 0 and 1 always produces decay. The boundary is exactly b = 1. With b = 1.5, every step up in x multiplies the output by 1.5, so the curve rises. With b = 0.6, every step multiplies by 0.6, so the curve falls toward zero.
No other information is needed to classify the curve; the value of b alone determines the direction.
The base relates to a percentage growth rate through b = 1 + r, where r is the rate written as a decimal. A base of 1.05 is the same relationship as a 5% growth rate, and a base of 0.9 is the same as a 10% decline.
This is exactly the substitution the growth formula reference and the continuous growth calculator make when they express the same mathematics using x₀, r, and t instead of b and x.
Read an exponential graph by checking where it crosses x = 0 (always at y = 1, since b0 = 1 for any base) and which way it bends away from that point. A curve rising away from (0, 1) as x increases indicates b > 1; a curve falling toward the x-axis as x increases indicates 0 < b < 1.
The steepness near any point also tells a story: the curve is flattest close to x = 0 and grows visibly steeper the further x moves in the direction of growth, since each step compounds on the last.
Yes, the exponent x can be any real number, including negative values and fractions, as long as the base is positive. A negative exponent gives a reciprocal (b−2 = 1/b2), and a fractional exponent gives a root (b0.5 = √b).
This unrestricted domain is exactly why the base, not the exponent, carries the positivity requirement. The function still needs to return a real number for every possible x.