Exponential Functions: Growth, Decay and the Number e

1. Introduction

Linear functions add the same amount at every step. Exponential functions multiply by the same factor at every step, and that difference changes everything. A savings account that earns interest on its interest, a virus spreading through a population, a radioactive sample fading away, and a viral video all follow the exponential pattern: the bigger the quantity, the bigger the next change.

Exponential growth is famous for how quickly it escapes intuition. Fold a sheet of paper in half 42 times and, in theory, it would be thick enough to reach the Moon. Each fold doubles the thickness, and doubling 42 times means multiplying by more than four trillion. Exponential decay is just as dramatic in the other direction, shrinking a quantity toward zero without ever quite reaching it.

This article defines exponential functions, reviews the laws of exponents, compares growth and decay, introduces the special number e, and applies the ideas to money, medicine and science. We finish with how exponentials sit inside machine learning, from softmax to learning-rate schedules to the vanishing-gradient problem.

2. What is an exponential function?

An exponential function has the form f(x) = a · bˣ. The variable x is in the exponent, which is what separates it from a power function such as x², where the variable is the base. The number a is the starting value (the value at x = 0), and b is the growth factor, the number you multiply by for every step of 1 in x. We require b > 0 and b ≠ 1, since 1ˣ is just the constant 1.

For the exponential column, the ratio between neighbouring values is always 2, while for the linear column the difference is always 3. That is the fingerprint: constant difference means linear, constant ratio means exponential. For f(x) = 3 · 2ˣ, the values are 3, 6, 12, 24, and each is twice the one before.

3. Laws of exponents

Working with exponentials means working with exponents, and a few laws handle almost everything. Each law follows from the idea that an exponent counts repeated multiplication. For example, 2³ · 2⁴ is three twos times four twos, which is seven twos.

Notice that a negative exponent does not make the answer negative. It means "take the reciprocal." This is what lets exponential graphs extend to the left of zero, where they shrink toward the axis: 2⁻¹ = 1/2, 2⁻² = 1/4, 2⁻³ = 1/8.

4. Growth and decay

The size of the factor b decides what happens. If b > 1, the function grows. If 0 < b < 1, it decays. The speed depends on how far b is from 1: 3ˣ grows faster than 2ˣ, and (1/3)ˣ fades faster than (1/2)ˣ.

Real situations usually give a percentage rate, and the factor is b = 1 + r where r is the rate as a decimal (negative for decline). A 4 percent rise has b = 1.04, and a 15 percent fall has b = 0.85. A common slip is using the rate itself, 0.04, as the factor.

Two worked examples. A town of 5,000 people growing 4 percent a year follows P(t) = 5000 · 1.04ᵗ, which gives 5000 · 1.04¹⁰ ≈ 7,401 after 10 years. A car worth 20,000 dollars that loses 15 percent each year follows V(n) = 20000 · 0.85ⁿ, so it is worth about 12,283 dollars after 3 years.

No real quantity grows exponentially forever. A population runs out of food, an epidemic runs out of susceptible people, and a company runs out of customers. Real growth often starts out exponential and then flattens into an S-shaped logistic curve as limits take hold. Exponential models are best treated as accurate for the early stage, and as a warning about how quickly the limits can arrive.

5. Features of the graph

Every basic exponential y = bˣ shares a few features. It passes through (0, 1), because b⁰ = 1. Its domain is all real numbers, but its range is only the positive numbers, because a positive base raised to any power stays positive. And it has a horizontal asymptote at y = 0: the curve gets closer and closer to the x-axis on one side without touching it.

Shifting and flipping work as with any function. In y = a · bˣ⁻ʰ + k, the number h slides the curve right, k lifts it, and the asymptote moves with k to y = k. A negative a turns the curve upside down.

6. Exponential versus linear and polynomial

Exponentials do not just beat linear functions in the long run. They beat every polynomial. Compare x², a polynomial, with 2ˣ. At x = 3 the polynomial is ahead (9 against 8), the two tie at x = 4 (both 16), and from x = 5 onward the exponential leads and pulls away at a staggering rate.

This is why "exponential" has become a word for runaway change, and why a process that doubles every period can look harmless at first and overwhelming a few periods later. A useful trick for spotting exponential data is to plot it on a logarithmic axis, where it turns into a straight line.

7. The number e

One base is special enough to have its own symbol: e ≈ 2.71828. It appears when growth is compounded continuously. Imagine 100 percent interest on 1 dollar for a year. Paid once at the end, you get 2 dollars. Paid in two halves, you get (1 + 1/2)² = 2.25. Split the year into more and more pieces, and the result does not grow forever. It creeps toward a limit, e.

The function f(x) = eˣ is the natural exponential. Its defining property, proved in calculus, is that its slope at any point equals its height at that point. That makes eˣ the function that describes anything growing at a rate proportional to its current size, and it is why e appears in physics, finance, biology and statistics. Any exponential bˣ can be rewritten with base e as b ˣ = e^(x ln b), so eˣ is the standard form.

8. Compound interest

If you invest a principal P at annual rate r, compounded n times per year for t years, the balance is A = P(1 + r/n)ⁿᵗ. With continuous compounding it becomes A = Peʳᵗ. Simple interest, by contrast, pays only on the original amount: A = P(1 + rt), a linear function.

Take 1,000 dollars at 5 percent for 10 years. Compounded yearly, 1000(1.05)¹⁰ ≈ 1,628.89. The more often it compounds, the higher the balance, but with a ceiling set by continuous compounding at 1000e⁰·⁵ ≈ 1,648.72.

9. Doubling time and half-life

How long does it take a quantity to double? For growth at rate r per period, solving (1 + r)ᵗ = 2 gives t = ln 2 / ln(1 + r). A handy approximation is the rule of 72: the doubling time is about 72 / (r in percent).

The table shows why small rate differences matter. At 2 percent, money doubles in about 35 years. At 7 percent it doubles in about 10, so over 30 years it doubles three times and grows roughly eightfold, while the 2 percent account has not even doubled once.

Decay has the mirror concept: half-life, the time for half of a quantity to disappear. If the half-life is h, the amount left after time t is N = N₀ · (1/2)^(t/h). Carbon-14 has a half-life of about 5,730 years, which is why it is used to date ancient organic material.

A sample with 20 percent of its carbon-14 left is older than 2 half-lives (25 percent) but younger than 3 (12.5 percent). Solving 0.5^(t/5730) = 0.2 with logarithms gives about 13,305 years, which lands between the two.

10. Solving exponential equations

An exponential equation has the unknown in the exponent. When both sides can be written with the same base, the exponents must be equal, and the equation becomes an easy one. Solve 2ˣ = 32: since 32 = 2⁵, x = 5. Solve 3^(x + 1) = 81: since 81 = 3⁴, x + 1 = 4 and x = 3. Solve 4ˣ = 8: write both as powers of 2, 2^(2x) = 2³, so 2x = 3 and x = 1.5.

Some equations, such as 5ˣ = 20, cannot be rewritten with a common base. These need logarithms, the inverse of exponentials, which are the topic of the next article. The answer there is x = ln 20 / ln 5 ≈ 1.861, which fits since 5¹ = 5 and 5² = 25 bracket 20.

11. AI and machine learning

Exponentials are woven through machine learning. The most direct use is softmax, which turns a list of raw scores into probabilities. Because eˣ is always positive, applying it to any scores, even negative ones, gives positive numbers. Dividing each by the total makes them add to 1: pᵢ = eˢⁱ / Σ eˢʲ. For scores 2, 1 and 0, the exponentials are 7.39, 2.72 and 1.00, which sum to 11.11, so the probabilities are 0.665, 0.245 and 0.090.

The sigmoid, σ(z) = 1/(1 + e⁻ᶻ), is built from the same function. The exponential also drives learning-rate decay: many training schedules shrink the step size by a constant factor, ηₜ = η₀ · γᵗ. With η₀ = 0.1 and γ = 0.9, the rate halves roughly every 6.6 steps.

Exponentials also explain a famous training problem. Backpropagation multiplies one factor per layer. If each factor is about 0.5, the signal after 20 layers is 0.5²⁰ ≈ 0.000001, so early layers barely learn. This is the vanishing gradient. If each factor is about 1.5, the signal becomes 1.5²⁰ ≈ 3,325, an exploding gradient. Careful initialisation and activation choices aim to keep the factor near 1.

One more practical point: eˣ overflows quickly. A score of 1,000 would make e¹⁰⁰⁰ too large for a computer to store, so softmax implementations subtract the largest score first. This does not change the probabilities, because subtracting a constant multiplies every exponential by the same factor, which cancels when dividing.

12. Common mistakes

13. Practice problems

Try these before looking at the answers.

1. Evaluate f(x) = 3 · 2ˣ at x = 0, x = 3 and x = −1.
2. Does y = 5 · 0.8ˣ show growth or decay? By what percent per step?
3. Simplify (2⁵ · 2³) / 2⁶.
4. Solve 3ˣ = 81.
5. What is 2,000 dollars worth after 5 years at 6% compounded yearly?
6. A culture of 500 bacteria doubles every 3 hours. How many are there after 12 hours?
7. A sample has a half-life of 10 days. If you start with 80 g, how much is left after 30 days?
8. Find the softmax of the scores [1, 1].

Answers.

1. f(0) = 3, f(3) = 3 · 8 = 24, f(−1) = 3 · 1/2 = 1.5.
2. Decay, since 0.8 < 1. The factor 0.8 means a 20% drop per step.
3. 2⁸ / 2⁶ = 2² = 4.
4. 81 = 3⁴, so x = 4.
5. 2000 · 1.06⁵ ≈ 2,676.45, about 2,676 dollars.
6. 12 hours is 4 doublings, so 500 · 2⁴ = 8,000.
7. 30 days is 3 half-lives, so 80 · (1/2)³ = 10 g.
8. Equal scores give equal probabilities: e/(e + e) = 0.5 each, so [0.5, 0.5].

14. Conclusion

An exponential function multiplies by a constant factor at every step. If the factor exceeds 1 it grows, if it is between 0 and 1 it decays, and either way its graph passes through (0, a), has a horizontal asymptote at zero and never goes negative. Percentages translate into factors with b = 1 + r, and the laws of exponents let you simplify and solve.

The special number e underlies continuous growth, and exponentials appear in compound interest, doubling times, half-lives and, in machine learning, in softmax, sigmoid, learning-rate decay and vanishing or exploding gradients. When an equation cannot be solved by matching bases, you need the inverse operation, which is the subject of the next article: logarithmic functions.