Exponential Function

Q: Why is the number e special?

Because e^x is its own derivative — d/dx(e^x) = e^x. No other base has this property. That makes e the natural base for exponential functions in calculus, physics, and probability. e ≈ 2.71828 arises naturally as the limit (1 + 1/n)^n as n approaches infinity, which is the value of $1 invested at 100% interest compounded continuously for one year.

Q: What is the doubling time formula?

For continuous exponential growth with rate k, doubling time t_d = ln(2) / k ≈ 0.693 / k. The Rule of 72 is a mental-math approximation: doubling time in years ≈ 72 / (percentage growth rate). At 6% growth, money doubles in 12 years; at 10%, in 7.2 years.

Q: What is the half-life formula?

For exponential decay with rate λ, half-life t_(1/2) = ln(2) / λ ≈ 0.693 / λ. Carbon-14's half-life is 5,730 years (radiocarbon dating); caffeine's is about 5 hours in the human body; many drug doses are spaced based on the drug's half-life so blood concentration stays in the therapeutic range.

Q: What is the inverse of an exponential function?

The inverse of a^x is the logarithm log_a(x). For the natural exponential e^x, the inverse is the natural logarithm ln(x). They cancel: ln(e^x) = x and e^(ln x) = x. Exponentials turn addition into multiplication; logarithms reverse the transformation, turning multiplication into addition.

An exponential function is any function where the variable appears in the exponent: $ f(x) = a^x $. Despite the simple form, exponential functions describe almost every process where a quantity changes by a fixed proportion per unit time — compound interest, radioactive decay, population growth, viral infection spread, signal attenuation, capacitor charging. The natural exponential $ e^x $ sits at the center of calculus because it is its own derivative. This study note covers the definition, key properties, the special role of $ e $, and the difference between exponential growth and exponential decay.

Exponential growth and decay curves — Exponential growth (red, increasing) and exponential decay (blue, decreasing) — both curves pass through (0,1) and approach an asymptote on one side.

The Definition

An exponential function has the form:

$$ f(x) = a^x $$

where $ a > 0 $ and $ a \neq 1 $. The base $ a $ is a positive constant; the variable $ x $ is the exponent. If $ a = 1 $, the function reduces to the constant 1, which is not exponential. If $ a = 0 $, the function is undefined for negative exponents.

More generally, you’ll see $ f(x) = b \cdot a^x $ or $ f(x) = a^{kx} $, where $ b $ and $ k $ are constants. These are still exponential functions — $ b $ is the value at $ x = 0 $, and $ k $ controls how fast the function changes.

Growth vs Decay

The behavior of $ a^x $ depends entirely on whether the base $ a $ is greater than or less than 1.

If $ a > 1 $: the function increases as $ x $ increases. This is exponential growth. Examples: $ 2^x, 3^x, e^x, 10^x $. The function rises slowly at first, then accelerates.
If $ 0 < a < 1 $: the function decreases as $ x $ increases. This is exponential decay. Examples: $ (1/2)^x, 0.95^x, e^{-x} $. The function starts high and approaches zero as $ x $ grows.

Both types pass through the point $ (0, 1) $, because $ a^0 = 1 $ for any positive $ a $. Both have the x-axis as a horizontal asymptote on one side (right side for decay, left side for growth).

Key Properties

Exponential functions satisfy a small set of identities that show up everywhere.

Multiplicative shift: $ a^{x+y} = a^x \cdot a^y $. Adding exponents corresponds to multiplying values.
Division: $ a^{x-y} = a^x / a^y $.
Power of power: $ (a^x)^y = a^{xy} $.
Zero exponent: $ a^0 = 1 $ for any $ a > 0 $.
Negative exponent: $ a^{-x} = 1 / a^x $. A negative exponent flips the function: $ 2^{-x} = (1/2)^x $.
Inverse function: the inverse of $ a^x $ is the logarithm $ \log_a x $. They cancel: $ \log_a(a^x) = x $ and $ a^{\log_a x} = x $.

The Number e

Among all possible bases, $ e \approx 2.71828 $ is special. $ e $ is the unique base for which the derivative of $ a^x $ equals $ a^x $ itself:

$$ \dfrac{d}{dx} e^x = e^x $$

This single property makes $ e $ the standard base for exponential functions in calculus, physics, and probability. $ e $ is defined as the limit:

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

This limit arises naturally from continuous compounding. If you invest $1 at 100% annual interest compounded $ n $ times per year, the value after one year is $ (1 + 1/n)^n $. As compounding becomes continuous ($ n \to \infty $), the value approaches $ e $ dollars. Hence the ‘natural’ exponential.

$ e $ is irrational and transcendental — it cannot be expressed as a fraction or as a root of any polynomial with rational coefficients. It also has a clean series expansion:

$$ e^x = \sum_{n=0}^{\infty} \dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots $$

Exponential Growth in Practice

Anything that grows by a fixed percentage per unit time follows an exponential. The general form is:

$$ N(t) = N_0 e^{kt} $$

Where $ N_0 $ is the initial quantity, $ k $ is the continuous growth rate, and $ t $ is time. Doubling time $ t_d $ (time to double): $ t_d = \ln 2 / k \approx 0.693 / k $.

The famous Rule of 72: doubling time in years ≈ 72 / (percent growth rate). At 6% annual growth, money doubles in 12 years. At 10%, in 7.2 years. The Rule of 72 is the Rule of 70 dressed up — 0.693 (the exact constant) rounded to 0.72 for easier mental math with the percentages that show up in finance.

Exponential Decay in Practice

Decay follows the same form with a negative exponent:

$$ N(t) = N_0 e^{-\lambda t} $$

Where $ \lambda > 0 $ is the decay constant. Half-life $ t_{1/2} = \ln 2 / \lambda $ — the time for the quantity to fall by half. Carbon-14 has half-life 5,730 years (radiocarbon dating). Caffeine has half-life ~5 hours (which is why coffee at 3pm still affects sleep). RC circuit voltage decays with a time constant $ \tau = RC $ according to $ V(t) = V_0 e^{-t/\tau} $.

Frequently Asked Questions

What is an exponential function?

An exponential function has the form f(x) = a^x, where a > 0 and a ≠ 1, and x is the variable in the exponent. The base a is constant; the variable appears in the exponent. Examples: 2^x, e^x, (1/2)^x, 10^x. Exponential functions describe processes that change by a fixed proportion per unit time.

What is the difference between exponential growth and decay?

If the base a > 1, the function grows as x increases (exponential growth, like 2^x). If 0 < a 0 and decay if k < 0. Both curves pass through (0, 1).

Why is the number e special?

Because e^x is its own derivative — d/dx(e^x) = e^x. No other base has this property. That makes e the natural base for exponential functions in calculus, physics, and probability. e ≈ 2.71828 arises naturally as the limit (1 + 1/n)^n as n approaches infinity, which is the value of $1 invested at 100% interest compounded continuously for one year.

What is the doubling time formula?

For continuous exponential growth with rate k, doubling time t_d = ln(2) / k ≈ 0.693 / k. The Rule of 72 is a mental-math approximation: doubling time in years ≈ 72 / (percentage growth rate). At 6% growth, money doubles in 12 years; at 10%, in 7.2 years.

What is the half-life formula?

For exponential decay with rate λ, half-life t_(1/2) = ln(2) / λ ≈ 0.693 / λ. Carbon-14’s half-life is 5,730 years (radiocarbon dating); caffeine’s is about 5 hours in the human body; many drug doses are spaced based on the drug’s half-life so blood concentration stays in the therapeutic range.

What is the inverse of an exponential function?

The inverse of a^x is the logarithm log_a(x). For the natural exponential e^x, the inverse is the natural logarithm ln(x). They cancel: ln(e^x) = x and e^(ln x) = x. Exponentials turn addition into multiplication; logarithms reverse the transformation, turning multiplication into addition.