From a grain of rice on a chessboard to a bank account earning continuous interest — exponential functions are everywhere growth is unbounded.
Why This Section Matters
The Most Powerful Pattern in All of Mathematics
The Chess Legend
When the inventor of chess presented his game to the king of India, the king was so impressed he offered any reward. The inventor asked for something seemingly modest: one grain of rice on the first square, two on the second, four on the third, doubling each time across all 64 squares.
The king laughed — then called his mathematicians. The total? $2^{64} - 1 \approx 1.8 \times 10^{19}$ grains — more rice than has ever been produced in human history. The king had no choice but to hand over his kingdom.
This is exponential growth. Each step multiplies by the same factor. The result quickly dwarfs anything we can intuitively grasp.
Folding Paper to the Moon
A standard sheet of paper is about 0.1 mm thick. If you could fold it in half 42 times, the stack would reach from the Earth to the Moon (384,400 km). If you fold it 103 times, it would reach the edge of the observable universe.
This is because each fold doubles the thickness: after $n$ folds, thickness $= 0.1 \times 2^n$ mm.
🚀 Try the Paper Folding Activity — Coming Soon
In this section: We make exponential growth precise, explore the special base $e$, and apply these ideas to the most important financial formula you will ever use — the continuous compounding formula $B = Pe^{rt}$.
§ 1 — The Exponential Function
Defining Exponential Functions
You already know functions like $f(x) = x^2$ or $g(x) = x^3$, where the variable is the base and the power is fixed. An exponential function reverses this: the base is a fixed positive constant, and the variable $x$ is the exponent.
Definition — Exponential Function
An exponential function is a function of the form
$$f(x) = b^x$$
where $b$ is a positive constant called the base, with $b > 0$ and $b \neq 1$, and $x$ is any real number.
The domain is all real numbers $(-\infty, \infty)$. The range is $(0, \infty)$.
Why can't b be negative or equal to 1?
If b = 1
$f(x) = 1^x = 1$ for every $x$. This is just the constant function $y=1$, a horizontal line. It has no interesting exponential behaviour and is trivially excluded.
If b < 0
$(-2)^x$ is undefined for irrational $x$ like $x = \sqrt{2}$. The function would not be defined on all of $\mathbb{R}$ and would not be continuous, useless for calculus.
Key distinction: In $f(x) = x^2$, the variable is the base and the exponent is fixed — this is a power function. In $f(x) = 2^x$, the variable is the exponent and the base is fixed — this is an exponential function. They behave completely differently.
The denominator is the root, numerator is the power.
$8^{2/3} = (\sqrt[3]{8})^2 = 4$
Extending to irrational exponents: What does $2^{\sqrt{2}}$ mean? We approximate $\sqrt{2} \approx 1.41421...$ and define $2^{\sqrt{2}} = \lim_{r\to\sqrt{2}} 2^r$ where $r$ takes rational values approaching $\sqrt{2}$. This limit exists and equals approximately $2.665$. This is how exponential functions are extended to all real numbers.
§ 3 — Graphing Exponential Functions
How the Graph Changes with the Base
The interactive explorer below lets you slide the base $b$ and see the graph change in real time. Pay attention to what happens as $b$ crosses 1.
📈 Exponential Graph Explorer — f(x) = bˣ
Base: b = 2↗ increasing (b > 1)
b = 0.1← decreasingb = 1increasing →b = 9
Domain
All real numbers (−∞, ∞)
Range
(0, ∞) — always positive
y-intercept
Always (0, 1) — any base
💡 Watch as you slide: When b crosses 1, the graph flips from decreasing to increasing. At b = 1 the graph is the horizontal line y = 1. The y-intercept is always (0, 1) regardless of b.
Two Families of Exponential Graphs
b > 1: Exponential Growth
The function $f(x) = b^x$ is increasing. As $x\to\infty$, $f(x)\to\infty$. As $x\to-\infty$, $f(x)\to 0$ (but never reaches 0). The x-axis is a horizontal asymptote on the left.
Examples: $2^x,\; 3^x,\; 10^x,\; e^x$
0 < b < 1: Exponential Decay
The function $f(x) = b^x$ is decreasing. As $x\to\infty$, $f(x)\to 0$. As $x\to-\infty$, $f(x)\to\infty$. The x-axis is a horizontal asymptote on the right.
Examples: $(1/2)^x,\; (0.8)^x,\; (1/3)^x$
Note: $(1/2)^x = 2^{-x}$. Replacing $x$ with $-x$ reflects the graph across the y-axis. A decreasing exponential is simply a growing exponential reflected left-right.
Example 1 — Sketch $f(x) = 3^x$ and $g(x) = (1/3)^x$
Key values table:
x
−2
−1
0
1
2
3
3ˣ
1/9
1/3
1
3
9
27
(1/3)ˣ
9
3
1
1/3
1/9
1/27
Both pass through $(0,1)$. They are reflections of each other across the y-axis since $(1/3)^x = 3^{-x}$. $3^x$ grows steeply right; $(1/3)^x$ decays to 0 right.
§ 4 — Properties of Exponential Functions
Key Properties of $f(x) = b^x$
Properties Table
Property
b > 1 (Growth)
0 < b < 1 (Decay)
Domain
$(-\infty,\infty)$
$(-\infty,\infty)$
Range
$(0,\infty)$
$(0,\infty)$
y-intercept
$(0,1)$
$(0,1)$
x-intercept
None
None
Behaviour
Increasing ↗
Decreasing ↘
As $x\to +\infty$
$f(x)\to +\infty$
$f(x)\to 0$
As $x\to -\infty$
$f(x)\to 0$
$f(x)\to +\infty$
Horizontal asymptote
$y=0$ (left side)
$y=0$ (right side)
One-to-one?
Yes
Yes
Example 2 — Using properties to solve $3^x = 3^{2x-1}$
Since $f(x) = 3^x$ is one-to-one, equal outputs mean equal inputs:
Why b ≠ 1? If $b=1$, then $1^x = 1^y = 1$ for every $x$ and $y$ — equal outputs tell you nothing about the inputs. The equality rule works only because $f(x)=b^x$ is one-to-one when $b\neq 1$: different inputs always produce different outputs, so $b^x = b^y$ can only happen when $x=y$.
Common Confusion — $b^x$ vs $x^b$: $f(x) = 2^x$: exponential — grows explosively as $x\to\infty$. $g(x) = x^2$: power function — grows polynomially as $x\to\infty$. For large $x$: $2^x \gg x^2$. In fact $\lim_{x\to\infty}\frac{2^x}{x^{100}} = \infty$ — exponential always wins over any polynomial, no matter the degree.
§ 6 — Solving Exponential Equations
Solving Equations Involving Exponents
Strategy: If you can write both sides with the same base, set the exponents equal (using the one-to-one property). If you cannot, use logarithms (covered in §4.2).
Among all possible bases for an exponential function, one is special: the irrational number $e \approx 2.71828...$ It arises naturally in calculus, finance, biology, and physics — anywhere continuous growth appears.
As $n$ grows, the expression approaches $e$ from below — never quite reaching it, but getting arbitrarily close.
Numerical Approximation
n
10
100
1,000
10,000
100,000
1,000,000
$(1+1/n)^n$
2.59374
2.70481
2.71692
2.71815
2.71827
2.71828
See §A.3: We proved this limit equals $e$ using L'Hôpital's rule — the $1^\infty$ indeterminate form. The key step was $\ln L = \lim_{x\to\infty} x\ln(1+1/x) = 1$, giving $L=e$. Review that derivation →
The Natural Exponential Function
$$f(x) = e^x$$
This is the exponential function. Its base $e \approx 2.71828$ is chosen precisely because $\frac{d}{dx}[e^x] = e^x$ — the only function that is its own derivative. We explore this in §4.3.
§ 8 — Continuous Compounding of Interest
Where Exponentials Meet Money
Suppose you invest a principal $P$ at an annual interest rate $r$ (as a decimal). How the interest is compounded — annually, monthly, daily, or continuously — determines how fast your money grows.
Step 1 — Compounded $k$ times per year: Each compounding period, the balance is multiplied by $(1 + r/k)$. After $T$ years (which is $kT$ periods):
$$B = P\left(1+\frac{r}{k}\right)^{kT}$$
Step 2 — What happens as $k\to\infty$? Let $n = k/r$, so $k = nr$:
Step 3 — Take the limit: As $k\to\infty$, $n\to\infty$, and we recognise the definition of $e$:
$$B = P\cdot e^{rT}$$
This is the continuous compounding formula. The number $e$ enters finance because continuous compounding is the mathematical limit of compounding infinitely often. ∎
Future Value of an Investment
Compounded k times/year
$$B = P\left(1+\frac{r}{k}\right)^{kT}$$
$P$ = principal · $r$ = annual rate $k$ = compounds/year · $T$ = years
Continuous Compounding
$$B = Pe^{rT}$$
$P$ = principal · $r$ = annual rate $T$ = years · $e \approx 2.71828$
💰 Compounding Explorer
Principal: PKR 1,000
Annual Rate: 8%
Years: 10 yr
Annual
PKR 2,159
+116%
Monthly
PKR 2,220
+122%
Daily
PKR 2,225
+123%
Continuous
PKR 2,226
+123%
Continuous compounding always gives the highest return — the mathematical upper bound for any rate r.
Example 8
Comparing Compounding Frequencies
PKR 50,000 is invested at 6% annual interest. Find the future value after 5 years under: (a) annual, (b) monthly, (c) continuous compounding.
Continuous compounding gives the highest return, but the difference from monthly is only PKR 50 — the gain from compounding more frequently diminishes rapidly.
Example 9
Doubling Time
How long does it take for an investment to double at 8% continuous compounding?
Suppose you want to have PKR $B$ in your account after $T$ years. How much should you invest today? This is the present value — the current worth of a future amount, discounted at the prevailing interest rate.
Present Value of an Investment
Compounded k times/year
$$P = B\left(1+\frac{r}{k}\right)^{-kT}$$
Continuous Compounding
$$P = Be^{-rT}$$
Present value is obtained by solving the future value formula for $P$. The factor $e^{-rT}$ (or $(1+r/k)^{-kT}$) is called the discount factor.
Example 11 — Planning a Future Expense
A LUMS student wants to have PKR 500,000 available in 4 years for graduate school. How much should they deposit today at 7% compounded continuously?
Depositing PKR 377,900 today at 7% continuously compounded will grow to PKR 500,000 in 4 years.
Example 12 — Comparing Two Investment Offers
Investment A pays PKR 800,000 in 5 years. Investment B pays PKR 650,000 in 3 years. Which is worth more today if the rate is 6% compounded continuously?
$P_A > P_B$, so Investment A is worth more today despite being received later. Present value is the correct tool for comparing payments at different times.
§ 10 — Effective Interest Rate
Which Investment Actually Pays More?
When two investments quote different rates and different compounding frequencies, you cannot compare them directly. The effective annual interest rate $r_e$ converts any investment to an equivalent annually compounded rate for fair comparison.
Effective Annual Rate
Compounded k times/year
$$r_e = \left(1+\frac{r}{k}\right)^k - 1$$
Continuous Compounding
$$r_e = e^r - 1$$
The investment with the higher $r_e$ is the better choice, regardless of how the nominal rates and frequencies differ.
Example 13
Bank A offers 8% compounded quarterly. Bank B offers 7.9% compounded continuously. Which is better?
Ranking: $\text{B} > \text{C} > \text{A}$. Despite B having the lowest nominal rate, its monthly compounding makes it the best choice.
Coming up next — §4.2 Logarithmic Functions — we introduce $\log_b x$ as the inverse of $b^x$, learn the logarithm laws, and solve exponential equations that cannot be solved by matching bases.