Exponential Functions

Part 14, Chapter 14: Functions and Their Graphs

Learning objectives

Define the exponential function $a^x$ and understand its properties
Solve basic exponential equations
Describe growth and decay behavior

Exponentials describe everything that doubles. Bacterial populations, compound interest, viral infections, radioactive decay, the cooling of coffee, the spread of rumours, whenever the rate of change is proportional to the amount present, the answer is an exponential. The function $b^x$ is so important that calculus invents a special base $e$ just so its derivative looks clean.

Definition and laws of exponents

For a positive real number $b \neq 1$ , the exponential function with base $b$ is $f(x) = b^x$ . The laws of exponents are not new, they extend the integer rules you already know:

$b^{x + y} = b^x \cdot b^y, \quad b^{x - y} = \dfrac{b^x}{b^y}, \quad (b^x)^y = b^{xy}, \quad b^0 = 1.$

The miracle is that the same identities hold for any real exponent, not just integers. Defining $b^{\pi}$ rigorously requires a limit; for now, take the laws on faith and use them.

Growth vs decay

If $b > 1$ , the function $b^x$ is strictly increasing: it grows without bound as $x \to +\infty$ and approaches $0$ as $x \to -\infty$ . This is exponential growth. If $0 < b < 1$ , the function is strictly decreasing, exponential decay. The two cases are mirror images: $2^x$ and $(1/2)^x = 2^{-x}$ are reflections of each other across the $y$ -axis.

Key features

Domain: all real numbers $(-\infty, \infty)$ .
Range: $(0, \infty)$ , an exponential is always positive.
The graph passes through $(0, 1)$ because $b^0 = 1$ .
The $x$ -axis is a horizontal asymptote, the curve hugs it but never touches.

The natural base $e$

Among all bases there is one distinguished choice, Euler's number

$e = \lim_{n \to \infty} \left(1 + \dfrac{1}{n}\right)^n \approx 2.71828\ldots$ ntoinftyleft(1+dfrac1nright)napprox2.71828ldots

The function $e^x$ is uniquely characterised by the property that it equals its own derivative: $\dfrac{d}{dx} e^x = e^x$ . That property, combined with the law $e^{x+y} = e^x e^y$ , is the reason $e^x$ is the natural language of calculus and differential equations.

Where this shows up

Biology: Population growth (E. coli in a Petri dish, viral case counts in an outbreak) follows $N(t) = N_0 e^{rt}$ until resources run low; $r$ is the growth rate, and doubling time is $\ln 2 / r$ .
Finance: Continuously compounded interest: $A = P e^{rt}$ . Every fixed-income pricing model on Wall Street uses the exponential function as its core building block.
Physics: Radioactive decay is $N(t) = N_0 e^{-\lambda t}$ ; carbon-14 dating uses the half-life of ${}^{14}C$ (about 5,730 years) plugged into this formula to date archaeological samples.

(The widget plots $b^x$ in one colour and $\log_b(x)$ b(x) in another. Slide the base $b$ and watch both curves morph together. The dashed diagonal is $y = x$ , the curves reflect across it because logarithm is the inverse of exponential.)

Solving exponential equations

To solve $2^x = 32$ , express both sides with the same base: $32 = 2^5$ , so $x = 5$ . More generally, the one-to-one property of exponentials says: if $b^x = b^y$ then $x = y$ . For trickier cases like $8^x = 2^{12}$ , rewrite the bases: $(2^3)^x = 2^{12}$ , so $2^{3x} = 2^{12}$ , giving $x = 4$ .

Try it

Predict first: what does $y = 1^x$ look like for any $x$ ? Slide the base to $b = 1$ in the widget, observe the curve, and explain why.
Predict first: how should the curve $y = 0.5^x$ relate to $y = 2^x$ ? Set $b = 0.5$ and describe in one sentence what you observe.
Solve $9^x = 27$ by hand. Check your answer by evaluating mentally; then check with the widget by reading off the appropriate intersection.

Pause: why must $b$ be positive and not equal to $1$ ? What goes wrong if $b = 0$ ? If $b = 1$ ? If $b = -1$ ?

Try it in code

A trap to watch for

Students see two exponents in a sum and try to add them. $e^{a+b} \neq e^a + e^b$ . The correct identity is $e^{a+b} = e^a \cdot e^b$ , sum in the exponent becomes product of values, not sum of values. Sanity check with small numbers: $e^{1 + 1} = e^2 \approx 7.39$ , while $e^1 + e^1 = 2e \approx 5.44$ . These are not equal. The reason the trap is sticky is that addition feels distributive over everything in arithmetic; with exponentials, addition is in the exponent, and the rule changes. Whenever you see an exponential, ask: is the operation outside the exponent linear? Or is it the law $b^{a+b} = b^a b^b$ ?

What you now know

You can recognise an exponential function by its formula and its graph, apply the laws of exponents to simplify expressions, solve simple exponential equations by base-matching, and explain why the natural base $e$ is distinguished. The next section introduces the function that inverts the exponential, the logarithm, and shows you the three identities that turn complicated products into simple sums.

Quick check

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References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 13 §4, exponential function definition and basic identities.
Stewart, J. (2015). Calculus: Early Transcendentals, 8th ed. Cengage. Sections 1.5 and 6.2, exponentials, growth/decay, and the natural base.
Maor, E. (1994). e: The Story of a Number. Princeton University Press. Historical and conceptual account of $e$ and the exponential function.