Everything in this section so far has treated the exponent as a known quantity — a specific number, whether natural, negative, rational, or irrational. The base was computed and the exponent told you what to do with it. Exponential functions reverse that relationship: the base is fixed and the exponent roams freely across all real numbers, turning a single arithmetic operation into a function with a distinctive shape and remarkable properties.
The Conceptual Shift
The expression 25 is a computation — it takes a fixed base and a fixed exponent and produces the number 32. The expression x2 is a polynomial — the base varies while the exponent stays at 2.
The expression 2x is something different. The base is locked at 2 and the exponent x is free to be any real number. As x changes, 2x traces out a curve: 20=1, 21=2, 22=4, 23=8, 2−1=21, 21/2=2.
This is the defining feature of an exponential function: a constant base raised to a variable exponent. The function f(x)=ax, with a>0 and a=1, assigns to every real number x a positive output determined by the laws of exponents.
The shift from "evaluate an for a specific n" to "study ax as x ranges over all reals" is the transition from arithmetic to function behavior — from individual calculations to a complete curve.
Basic Shape
The graph of f(x)=ax takes one of two forms, depending on whether the base is greater than 1 or between 0 and 1.
When a>1, the function grows. For large negative values of x, ax is close to zero — positive but tiny. At x=0, the function passes through (0,1) because a0=1. As x increases, ax rises with accelerating steepness. The curve climbs slowly at first, then explosively.
When 0<a<1, the function decays. The curve is a mirror image — high on the left, passing through (0,1), and falling toward zero on the right. Each step to the right multiplies by a fraction, shrinking the output.
The point (0,1) lies on every exponential graph, regardless of the base. This is a direct consequence of the zero exponent rule: a0=1 for any positive a.
The larger the base (when a>1), the steeper the growth. The function 10x rises far more aggressively than 2x. The closer the base is to 1 from either side, the flatter the curve — 1.01x grows, but barely.
Key Properties
Every exponential function f(x)=ax with a>0 and a=1 shares the same structural properties.
The domain is all real numbers. Every real x — positive, negative, zero, rational, irrational — produces a well-defined output, because the base is positive.
The range is (0,∞). The output is always positive — never zero, never negative. No matter how far left the curve extends, it approaches the horizontal axis but never reaches it.
The horizontal asymptote is the line y=0. For a>1, the curve approaches zero as x→−∞. For 0<a<1, it approaches zero as x→+∞. In neither case does the function touch the axis.
The function is one-to-one. When a>1, it is strictly increasing — different inputs always produce different outputs. When 0<a<1, it is strictly decreasing. This is the property that makes exponential equations and inequalities solvable: ax=ay implies x=y.
There is no x-intercept. Since ax>0 for all x, the graph never crosses the horizontal axis.
Exponential vs Power Functions
The expressions x2 and 2x look similar on paper but behave in fundamentally different ways. In x2, the variable is the base — this is a power function, a polynomial. In 2x, the variable is the exponent — this is an exponential function.
For small positive values of x, the polynomial can dominate. At x=2, x2=4 while 2x=4 — they are equal. At x=3, x2=9 while 2x=8 — the polynomial is still ahead.
But exponential growth eventually overtakes any polynomial, no matter the degree. At x=10, x2=100 while 2x=1024. At x=20, x2=400 while 2x=1,048,576. The gap does not just widen — it accelerates.
This holds for polynomials of any degree. The function 2x eventually surpasses x10, x100, even x1000. Exponential growth multiplies by a fixed factor at each step, while polynomial growth adds a fixed power. Repeated multiplication always wins in the long run.
Euler's Number e
Among all possible bases for an exponential function, one holds a privileged position: the irrational number e≈2.71828.
The function f(x)=ex is called the natural exponential function. Its defining property is that the rate at which it grows at any point equals the value of the function at that point. At x=0, the function equals 1 and is growing at rate 1. At x=1, the function equals e≈2.718 and is growing at rate ≈2.718. The output and the growth rate are always identical.
No other base has this property. The function 2x grows at a rate proportional to its value, but not equal to it — a correction factor is needed. The function 3x overshoots. Only ex achieves exact self-replication of value and rate.
This property makes ex central to calculus, differential equations, compound interest calculations at continuous rates, and mathematical modeling across the sciences. The full development of why e takes this value and what follows from it belongs to those subjects — but its origin lies here, in the extension of exponentiation to all real numbers.
Where to Go Next
This page marks the boundary of the powers section and the beginning of several deeper topics that build on exponential functions.
Logarithms are the inverse operation of exponentiation. If ax=b, then x=loga(b). Every law of exponents has a corresponding logarithmic identity, and the two subjects are inseparable. Logarithms will be covered in their own dedicated section.
Transformations of exponential functions — shifts, reflections, stretches — modify the basic curve ax into forms like 3⋅2x−1+5. These belong to the broader study of function transformations.
Calculus of exponential functions — derivatives and integrals of ax and ex — represents one of the most elegant chapters in mathematics, where the self-replicating property of ex reaches its full expression.
Applications draw on all of the above. Compound interest, population growth, radioactive decay, cooling processes, and probability distributions all rest on exponential functions — and on the exponent framework developed across this entire section, from the first definition of an as repeated multiplication to the continuous curve of ax for all real x.