The seven numbers \tfrac{1}{8}, \tfrac{1}{4}, \tfrac{1}{2}, 1, 2, 4, 8 are the most important values of 2^x you will meet in all of school mathematics. They appear in binary place value, in compound interest approximations, in biology's doubling times, in Diwali-size firecracker calculations, in the powers-of-two that underlie every computer memory module. They are worth knowing in your bones.

The picture below lets you feel them as points on a single smooth curve, not as disconnected table entries. Drag the red point along the exponent axis. The value of 2^x updates in real time, and you will see those seven landmark values flash past as you sweep from -3 to +3.

The picture

Exponent slider showing the curve y equals two to the power of x with a draggable pointThe curve y equals two to the power of x is plotted over the exponent range from minus three to plus three. The y-axis runs from zero to eight. A draggable red point sits on the curve, and the reader can drag it left or right along the exponent axis. Readouts at the top show the current exponent x, the corresponding value two to the x, and the fraction or integer form. Seven dashed horizontal lines mark the landmark values one-eighth, one-quarter, one-half, one, two, four, and eight. x y 8 4 2 1 1/2 1/4 -3 -2 -1 0 1 2 3 ↔ drag the red point along the curve
Drag the red point along the curve $y = 2^x$. As $x$ sweeps from $-3$ to $+3$, the value of $2^x$ sweeps through $\tfrac{1}{8}, \tfrac{1}{4}, \tfrac{1}{2}, 1, 2, 4, 8$ in perfect order. The curve is smooth — no gaps, no jumps — which is why $2^x$ is defined for *every* real $x$, not just integers.

Why the curve has no jumps: y = 2^x is a continuous function. Between any two values of x, all intermediate values of y are visited. So at some exponent between 0 and 1, the curve passes through y = 1.5; at some exponent between 1 and 2, it passes through y = 3. The whole point of the picture is that 2^x is not a sparse table — it is a continuous, ever-rising curve that happens to hit nice values at the integer exponents.

The seven landmark values

Drag the point to each of these seven positions. Each integer exponent gives a landmark value, and the relationship between consecutive values is always "multiply by 2" (going right) or "divide by 2" (going left).

Memorising these seven values — plus the rule that each step of 1 in x doubles the value — gives you a running head start on everything from binary arithmetic to scientific notation.

What the shape reveals

Three features of the curve are worth noticing as you drag.

The curve never touches the x-axis. As you drag the point to the far left, y gets smaller and smaller — \tfrac{1}{16}, \tfrac{1}{32}, \tfrac{1}{1024} — but it never becomes zero. This matches the rule 2^x > 0 for every real x: there is no finite exponent that makes 2^x disappear. The x-axis is an asymptote, and the curve approaches it without crossing.

The curve is exactly 1 at x = 0. Every exponential curve y = a^x with a > 0 passes through the point (0, 1), because a^0 = 1. That is the single anchor point all exponential curves share.

Each step of 1 in x multiplies y by the base. Go from x = 0 to x = 1 — the value doubles, 1 \to 2. Go from x = 1 to x = 2 — it doubles again, 2 \to 4. Go from x = 0 to x = -1 — it halves, 1 \to \tfrac{1}{2}. This rule, "step right = multiply by base, step left = divide by base," is the geometric heart of the exponent laws. It is why 2^m \times 2^n = 2^{m + n}: you are combining two steps on the exponent axis, and the total multiplication on the y-axis is the product.

Why negative exponents are reciprocals: stepping from x = 0 to x = -1 divides y by 2. Starting from y = 1 (at x = 0), one step left gives y = \tfrac{1}{2}. Two steps left gives y = \tfrac{1}{4}. So "negative exponent" just means "number of divisions by the base, starting from 1." The formula 2^{-n} = \tfrac{1}{2^n} is the bookkeeping for that counting.

Between the integers, what happens?

Drag the point to fractional exponents like x = 0.5 and x = 1.5. At x = 0.5, the readout shows about 1.414 — which is \sqrt{2}. At x = 1.5, the readout shows about 2.828 — which is 2\sqrt{2}. Fractional exponents are roots, built into the same curve: 2^{1/2} is the number which, when squared, gives 2. The smoothness of the curve is the geometric promise that such a number exists, for every fractional exponent.

Irrational exponents like x = \pi or x = \sqrt{3} are harder to picture, but the curve handles them too — it is continuous, so every real x has a well-defined value of 2^x. This is explained in more depth in the Exponents and Powers chapter.

The carry-away

Every time you encounter a number like 64 = 2^6 or 0.015625 = 2^{-6} or the fact that "a doubling every hour for three hours gives an 8 \times multiplication," you are using this same curve. The slider above is the picture you should carry in your head whenever an exponent problem starts. The seven landmark values tell you where you are; the smooth curve tells you that the rules that work at integer exponents — doubling, halving, multiplying — still apply in between.

Related: Exponents and Powers · Percentages and Ratios · Number Systems · Roots and Radicals