Exponential Function

Summary: An exponential function $a^x$ is one where the input lives in the exponent; its defining identity $a^{x+y}=a^x\cdot a^y$ is what forces every exponential to be proportional to its own derivative.

Key identity: $a^{x+y}=a^x\cdot a^y$: addition in the exponent (input) becomes multiplication in the output.

Definition

For a fixed base $a>0$, the exponential function is $f(x)=a^x$. The base is constant; the variable is the exponent. This is the opposite of a power function $x^a$, where the exponent is fixed and the base varies.

Interpretations of the definition

• For integer $n$, $a^n$ is repeated multiplication: $a\times a\times a\cdots$, $n$ times
• For exponent between 0 and 1, $a^{p/q}$ is a root specifically the $q$-th root of $a^p$.
  • Ex. 1: $a^{1/2}=\sqrt{a}$, the number which when squared gives $a$
  • Ex. 2: $a^{2/5}=\sqrt[5]{a^2}$, the number which when raised to the 5th power gives $a^2$
• For negative exponent, $a^{-n}$ is the reciprocal of $a^n$: $\frac{1}{a^n}$. Extending the repeated multiplication pattern “backwards” — each step divides by $a$ rather than multiplies
  • Ex. 1: $a^{-3}=\frac{1}{a^3}=\frac{1}{a\times a\times a}$, the reciprocal of $a$ multiplied by itself 3 times
  • Ex. 2: $a^{-2/5}=\frac{1}{\sqrt[5]{a^2}}$, the reciprocal of the 5th root of $a^2$
• For rational $p/q$, $a^{p/q}$ is the unique positive number satisfying $(a^{p/q}{)}^q=a^p$
  • Ex: $a^{3/2}$ is the unique positive number satisfying $(a^{3/2}{)}^2=a^3$
• For irrational and (later) complex inputs,
  • Ex. 1: $a^{\pi }$: since $\pi \approx 3.14159\dots$, we take the limit of $a^3,a^{3.1},a^{3.14},a^{3.141},\dots$ which converges to a unique real number
  • Ex. 2: $e^{i\pi }=-1$: complex exponent lands back on the real line; more generally $e^{i\theta }=\cos \theta +i\sin \theta$, so exponentiation by a complex number encodes rotation in the complex plane
  • The defining property below is what extends the function continuously

The key identity

$a^{x+y}=a^x\cdot a^y$

This is the only identity you need to remember for exponentials. It says: adding in the exponent (input) is multiplying in the output. It is the bridge between additive structure (steps in time) and multiplicative structure (rates, ratios, scaling).

Three corollaries fall straight out:

• $a^0=1$
  • Set $y=-x$ above, you need $a^0$ to cancel
• $a^{-x}=1/a^x$ (This one requires the $a^0=1$ corollary first — it chains on it.)
  • Set $y=-x$: $a^{x+(-x)}=a^x\cdot a^{-x}$,
    • so $a^0=a^x\cdot a^{-x}$,
    • so $1=a^x\cdot a^{-x}$,
    • giving $a^{-x}=\frac{1}{a^x}$.
• $(a^x{)}^y=a^{xy}$
  • Apply the identity repeatedly: $a^{xy}=a^{x+x+\cdots +x}$ ($y$ times) $=a^x\cdot a^x\cdots a^x=(a^x{)}^y$.
  • Clean for integer $y$; for the general case it follows from the continuous extension.

Strip everything else away and this identity is what makes something an exponential function.

Differentiating any exponential

The identity does almost all the work. Start from the definition:

$$\frac{d}{dt}{a}^t=\underset{dt\to 0}{\lim }\frac{a^{t+dt}-a^t}{dt}$$

Apply identity $a^{t+dt}=a^t\cdot a^{dt}$ and factor out $a^t$:

$$\frac{d}{dt}{a}^t=a^t\cdot \underset{k_a\text{: constant in }t}{\underbrace{\underset{dt\to 0}{\lim }\frac{a^{dt}-1}{dt}}}$$

The key observation: the limit on the right doesn’t depend on $t$ at all. Every exponential is therefore proportional (but, in almost all cases, not equal) to its own derivative, with a base-specific proportionality constant:

$$\frac{d}{dt}{a}^t=a^t\cdot k_a$$

$k_a$ can be computed numerically for any base, $a$, by substituting very small values for $dt$,

base $a$	constant $k_a$
2	$0.6931\dots$
3	$1.0986\dots$
$e\approx 2.718$	$1$
8	$2.0794\dots$ ($=3\cdot k_2$)

The constant $k_8=3k_2$ is the first hint that these constants behave logarithmically. They turn out to be exactly $\ln (a)$ — see natural-logarithm.

Why this matters

Whenever a system’s rate of change is proportional to the current quantity — population growth without resource limits, radioactive decay, Newton’s cooling, continuously compounded interest — the solution is an exponential. The proportionality property derived above is why: it’s the only family of functions that satisfies $\dot{f}\propto f$.

The “preferred” base for writing such solutions is $e$, because the proportionality constant becomes literally visible in the exponent. See eulers-number.

Sources

• What’s so special about Euler’s number e?