1
The variable is the exponent, not the base
x² and 2^x agree at x = 2 and at very little else. In x² the exponent is fixed and the base moves; in 2^x the base is fixed and the exponent moves. Only the second is an exponential function.
Lesson · Exponential and Logarithmic Functions · Exponential functions
Exponential Functions
In every function so far the variable sat in the base: x², x³, √x. An exponential function puts it in the exponent instead, and that one move changes what the values do — a constant ratio between steps, where a line has a constant difference.
f(x) = a·b^x a ≠ 0, b > 0, b ≠ 1
f(0) = a f(x + 1) / f(x) = b at every x
b^x > 0 for every real x ⟹ y = 0 is never reached
3·2^x : 3, 6, 12, 24, 48 ratios 2, 2, 2, 2
The definition, and what it forces
Core ideas
What this lesson establishes
Three things carry the lesson: where the variable sits, what the base is not allowed to be, and the one number that separates an exponential from a line.
2
A line adds, an exponential multiplies
Raise x by one and a linear function gains a fixed amount. An exponential is multiplied by a fixed amount. That multiplier is b, and it is the only thing needed to recognise one from a table.
3
The base is fenced in, and each fence excludes a real failure
b must be positive, or the function has no value at x = 1/2. b must not be 1, or the function is the constant a. Neither restriction is decoration; each removes something that genuinely breaks.
The material
The definition, and the two restrictions
What an exponential function is, why the base is fenced in, and what the graph therefore has to look like.
Definition · Exponential function
An exponential function is a function of the form f(x) = a·b^x with a ≠ 0, b > 0 and b ≠ 1. The number b is the base, and a is the value at x = 0.
Foundations of Algebra gave b^x a meaning for every rational x: whole exponents by repeated multiplication, negative ones by reciprocal, and b^(p/q) as the q-th root of b^p. For an irrational exponent — 2^√2 — the value is the number that the rational exponents close in on. This course takes that for granted and treats the domain as every real number.
Why the base must be positive
Take b = −4. Then b^1 = −4 and b^2 = 16 are perfectly good numbers, and so is b^(1/3) = −∛4. But b^(1/2) = √(−4) is not real, and neither is b^(1/4). What survives is a scattered set of rationals with no interval anywhere in it, so there is no curve to draw. Excluding a negative base is not squeamishness; it is the only way the function gets a domain worth having.
b = 0 fails differently: 0^x is 0 for every positive x, has no value at all for negative x, and 0^0 is a convention rather than something the laws force. And b = 1 does not fail at all — 1^x = 1 for every x — it simply gives the constant function f(x) = a, a horizontal line. It is excluded because calling that exponential would make every statement in “Growth and Decay” false.
Constant ratio, not constant difference
Theorem · The ratio is the base
For f(x) = a·b^x, the quotient f(x + 1) / f(x) equals b at every x. A linear function has g(x + 1) − g(x) constant instead. Given values at equally spaced inputs, computing both columns settles which kind of function you are looking at — if it is either.
f(x + 1) / f(x) = (a·b^(x+1)) / (a·b^x). Both a and b^x are non-zero, so the quotient is defined. The exponent law b^(x+1) = b^x·b reduces it to b, and b does not depend on x, which is the whole content of the claim.
□
x −3 −2 −1 0 1 2 3
2^x 1/8 1/4 1/2 1 2 4 8
every entry is positive, and the left end is getting small,
not getting to zero
The line the curve never touches
Every value of b^x is positive, so f(x) = a·b^x never takes the value 0. For b > 1, as x runs off to the left the values shrink without limit and the curve settles toward the horizontal line y = 0 without arriving. That line is the horizontal asymptote, and the reason the curve never reaches it is not that it runs out of room — it is that no real exponent makes a positive base give 0.
Definition · Horizontal asymptote
A horizontal line y = k is a horizontal asymptote of f when the values of f(x) get and stay arbitrarily close to k as x runs off to one end. For f(x) = a·b^x + k the asymptote is y = k: the power part shrinks toward 0, and k is what is left.
Example · Three formulas, read without drawing
f(x) = 2^x has a = 1, base 2, f(0) = 1, asymptote y = 0, and range every positive number. f(x) = 2^x + 3 is the same curve lifted by 3: asymptote y = 3, f(0) = 4, range every number above 3. f(x) = 3·(1/2)^x has f(0) = 3 and values 3, 3/2, 3/4, 3/8, 3/16 at x = 0, 1, 2, 3, 4 — still a constant ratio, and the ratio is 1/2.
Interactive
What makes a function exponential
A constant ratio, where a line has a constant difference
Method
Identifying an exponential function
Four checks. The first two settle what it is; the last two get the numbers out.
1
Find where the variable sits
Variable in the exponent with a fixed base: exponential. Variable in the base with a fixed exponent — x³, x^(1/2) — not exponential, whatever the exponent is. x^x has it in both places and is neither; it is outside this course.
2
Check the base against the two restrictions
Positive, and not 1. A negative base has no curve to draw, and b = 1 is a horizontal line wearing exponential notation.
3
Read a off at x = 0
b^0 = 1 for every allowed base, so f(0) = a. It is the fastest number to extract from a formula and the first one to test against a table.
4
Divide neighbouring values, do not subtract them
For values at equally spaced inputs, form the ratios. A constant ratio means exponential and names b; a constant difference means linear. Doing both costs one line and removes the guess.
Which function is which
Score 0 / 0Choose the answer you can justify.
One table, two functions
Both functions pass through (0, 3) and (1, 6). After that they have nothing in common, and the ratio column is where you see it first.
x 0 1 2 3 4
─────────────────────────────────────────────────
g(x) = 3 + 3x 3 6 9 12 15
difference 3 3 3 3 constant
ratio 2 3/2 4/3 5/4 not constant
f(x) = 3·2^x 3 6 12 24 48
difference 3 6 12 24 not constant
ratio 2 2 2 2 constant
so f is exponential with a = 3, b = 2
g is linear with intercept 3 and slope 3
At x = 4 the two are 48 and 15. They were equal three steps earlier, at x = 1, which is why two matching points prove nothing: a line and an exponential can always be made to agree at two points, and they part immediately afterwards.
The ratio row for g is worth reading. 2, 3/2, 4/3, 5/4 is not constant, and it is drifting toward 1 — a line eventually grows by a smaller and smaller proportion, while an exponential never does.
For a faded classification, use the table x: 0, 1, 2, 3 and h(x): −4, −12, −36, −108. The first supplied move is (−12)/(−4) = 3. Finish the ratio and difference rows, classify the table, and state a, b and the horizontal asymptote. Check the formula you obtain against all four values before opening the quiz.
Common mistakes
1
Reading 2^x as 2xAt x = 1 and x = 2 they agree — 2 and 4 — which is exactly why the habit survives past the first example. At x = 5 they are 32 and 10.
2
Treating a·b^x as (a·b)^x3·2^x at x = 2 is 3·4 = 12, while (3·2)^2 = 36. Order of operations does the power first: the coefficient multiplies the power and is not part of the base.
3
Saying the curve reaches its asymptote2^(−20) = 1/1048576. That is small and it is not 0, and there is no input further out that does better in kind. The asymptote is approached, never attained.
Check your understanding
✓
Finish when a table tells you the kind of function in one line.
Given values at equally spaced inputs you should form the ratios first, name b if they are constant, and state a. Given a formula you should state the base, f(0) and the horizontal asymptote without drawing anything.
Note: The restrictions b > 0 and b ≠ 1 are what make growth and decay, the number e, and the logarithm and its graph possible. A positive base other than 1 takes every positive value exactly once, which is what lets “What a Logarithm Is” define a logarithm as the exponent that produces a given number — the exponent, singular.