Exponential Functions

The variable belongs in the exponent

You have spent whole chapters on expressions like $x^2$, $x^3$, and general polynomials. In every one of them the variable sits in the base and a fixed number sits in the exponent. An exponential function turns that arrangement around.

A function is exponential when the variable appears in the exponent while the base is a fixed number. Its basic form is

where the constant $b$ is called the base. A slightly fuller form, $f(x) = a \cdot b^x$, includes a constant multiplier $a$ that sets the starting value; the plain form $b^x$ is just the case $a = 1$. Throughout the graphing below we take $a > 0$, so every output stays positive.

The whole idea lives in one comparison. Look at $2^x$ next to $x^2$:

They are built from the same two symbols, a $2$ and an $x$, and beginners read them as if they were the same thing. They are not even close. In $x^2$ the exponent is frozen at $2$ and the base $x$ moves; that is a power function. In $2^x$ the base is frozen at $2$ and the exponent $x$ moves; that is an exponential function. Watch how differently they grow:

The two happen to agree at $x = 2$ and again at $x = 4$, but their engines are different, and past $x = 4$ the exponential pulls away and never looks back. By $x = 10$ the power function reaches only $10^2 = 100$, while the exponential is already at $2^{10} = 1024$, more than ten times larger.

Which bases make sense

Not every number is allowed as a base. Two restrictions come straight from what $b^x$ has to mean if it is going to be a function defined for every input $x$.

The base must be positive. Suppose you tried a negative base such as $b = -4$. As long as the exponent is a whole number you get an answer, but the sign flips back and forth: $(-4)^1 = -4$, $(-4)^2 = 16$, $(-4)^3 = -64$. Worse, a fractional exponent asks for a root of a negative number, and

is not a real number at all. So a negative base leaves the function undefined at many inputs and lurching in sign at the rest, which is no way to build a smooth curve. The base $b = 0$ fails too, since $0^x$ is $0$ for positive $x$ and undefined for $x \le 0$. Ruling these out, we require $b > 0$.

The base must not be $1$. As the worked example showed, $1^x = 1$ for every $x$, so the “function” is just the constant $1$. It carries none of the growth or decay that makes an exponential interesting, so we set it aside.

Putting the two conditions together, an exponential function $f(x) = b^x$ uses a base with

From whole-number powers to a smooth curve

The definition $b^x$ has to make sense for every real input, not just whole numbers, before we can draw it as an unbroken curve. Everything needed is already in hand from earlier chapters on exponents:

• $b^0 = 1$ for any allowed base.
• A negative exponent gives a reciprocal, $b^{-n} = \dfrac{1}{b^n}$, which is a positive fraction, never a negative number.
• A unit-fraction exponent gives a root, $b^{1/n} = \sqrt[n]{b}$, and a general fraction combines the two, $b^{m/n} = \left(\sqrt[n]{b}\right)^m$.

Those rules pin down $b^x$ at every rational input. The remaining inputs, the irrational ones like $x = \sqrt{2}$, are filled in by the values crowding around them. For instance, $2^{\sqrt 2}$ sits snugly between $2^{1.41}$ and $2^{1.42}$, and squeezing the rational exponents closer traps it at a single value. We will not prove that squeezing step carefully here. Accept the outcome, which is that $b^x$ is defined for every real number $x$ and traces one smooth, unbroken curve that is positive everywhere. Here is the curve’s backbone for $b = 2$:

Two features of that row are worth naming now, because both trip people up later. The value at $x = 0$ is $1$, not $0$. And the values at negative inputs are small positive fractions, not negative numbers.

The rule that makes a function exponential

One identity separates exponential functions from every other kind, and it is the reason they behave the way they do. For an exponential function, adding inputs turns into multiplying outputs.

This turns hard-looking calculations into easy ones. Since $2^{10} = 1024$, you can find nearby powers without multiplying out a long string of $2$s. Adding $3$ in the exponent multiplies by $2^3 = 8$, and subtracting $3$ divides by $8$:

Equal steps, constant ratio

The step-by-step reading of the rule, “each unit added to $x$ multiplies the output by $b$,” gives a practical test for spotting exponential behavior in a table. Line up equally spaced inputs and look at what happens between neighbors. A linear function adds the same amount each step, a constant difference. An exponential function multiplies by the same amount each step, a constant ratio.

Compare a linear rule $g(x) = 3 + 5x$ with an exponential rule $f(x) = 3 \cdot 2^x$ over the same inputs:

For $g$, the jumps are $5, 5, 5, 5$, a constant difference, so successive values climb by the same amount. For $f$, the jumps are $3, 6, 12, 24$, which are not constant at all; instead the ratios $\tfrac{6}{3}, \tfrac{12}{6}, \tfrac{24}{12}, \tfrac{48}{24}$ are all $2$. Whenever equally spaced inputs produce a constant ratio between outputs, the function is exponential, and when those inputs step by $1$ that ratio is the base.

Growth, decay, and the shape of the graph

Whether an exponential rises or falls is decided entirely by the base.

When $b > 1$, each step to the right multiplies by a number bigger than $1$, so the outputs get larger. The function increases, and the picture is exponential growth.

When $0 < b < 1$, each step to the right multiplies by a number smaller than $1$, so the outputs shrink toward zero. The function decreases, and the picture is exponential decay. A decay curve is a mirror image of a growth curve, because raising one-half to the $x$ is the same as raising two to the negative $x$: $\left(\tfrac{1}{2}\right)^x = 2^{-x}$.

Both curves, growth and decay, share the same list of features, and reading them off the picture is the payoff of this lesson:

• Passes through $(0, 1)$. Every $b^x$ hits $1$ at $x = 0$, since $b^0 = 1$. For the fuller form $a \cdot b^x$ the $y$-intercept is $a$, because $a \cdot b^0 = a$.
• Always positive. A positive base raised to any real power is positive, so the graph lives entirely above the horizontal axis. The range is $y > 0$.
• Horizontal asymptote $y = 0$. On the flat side, growth to the left and decay to the right, the curve creeps toward the axis and gets arbitrarily close without ever touching it. The output is never $0$ and never negative.
• Domain is all real numbers. You may raise the base to any power, so every $x$ is allowed.
• One-to-one. The curve is strictly increasing (growth) or strictly decreasing (decay), so it never repeats an output. A one-to-one function has an inverse, and the inverse of an exponential function is the logarithm, the subject that closes this chapter. We only name it here; the tools for computing with it come later.

One base deserves a mention by name. The number $e \approx 2.718$ is a famous choice of base whose special role shows up in the next lesson, on compound interest. For now, treat it as just another base greater than $1$: its graph is a growth curve like the one above.

Common mistakes