1. What an exponential function is
A function of the form $f(x) = a \cdot b^x$, where the base $b > 0$ and $b \neq 1$, and the coefficient $a \neq 0$. The variable lives in the exponent, not the base.
The whole topic hinges on that last sentence. In $x^2$, the variable is the base and the exponent is fixed — that's a polynomial. In $2^x$, the variable is the exponent and the base is fixed — that's exponential. They look superficially similar; they behave nothing alike.
Polynomial vs exponential
- $2^x$ — base 2, variable in exponent
- $f(x) = 100 \cdot (1.05)^x$ — 5% growth
- $y = e^{-x}$ — natural decay
- $g(t) = 3 \cdot 2^{t/3}$ — doubles every 3
- $x^2$ — polynomial (variable in base)
- $x^x$ — not standard form
- $f(x) = 1^x = 1$ — constant, excluded
- $(-2)^x$ — base negative, not allowed over $\mathbb{R}$
The growth rule of $f(x) = b^x$ is the defining trait: every time $x$ increases by 1, $f$ is multiplied by $b$. That's a constant ratio per step — not a constant difference, which is what a linear function gives you. The whole shape of the function follows from that one rule.
Laws of exponents (reminder)
These hold for any allowed base $b$ and any real exponents $x, y$. They are not new — they're the same laws you used for integer exponents, extended.
$$ b^{x+y} = b^x \cdot b^y, \qquad b^{x-y} = \frac{b^x}{b^y}, \qquad (b^x)^y = b^{xy}, \qquad b^0 = 1, \qquad b^{-x} = \frac{1}{b^x}. $$2. Growth vs decay
The base $b$ alone tells you the qualitative behavior:
- $b > 1$ — exponential growth. The curve climbs ever faster as $x$ increases; for $x \to -\infty$ it flattens onto the x-axis.
- $0 < b < 1$ — exponential decay. The curve falls ever more slowly toward zero as $x \to \infty$; for $x \to -\infty$ it shoots up.
- $b = 1$ — excluded. $1^x = 1$ for every $x$, a horizontal line. Not interesting, and breaks the inverse.
Every exponential, regardless of base, shares structure:
- Domain: all real numbers.
- Range: $(0, \infty)$ — the function is always positive.
- Horizontal asymptote at $y = 0$ on one side ($x \to -\infty$ for growth, $x \to +\infty$ for decay).
- Passes through $(0, a)$ — at $x = 0$, $b^0 = 1$, so $f(0) = a$.
If a quantity grows by 6% per year, the multiplier per year is $1 + 0.06 = 1.06$, not $0.06$. The model is $P(t) = P_0 (1.06)^t$. Confusing factor with rate is the single most common modeling mistake.
Doubling time and half-life
Because the function multiplies by a fixed ratio each step, certain landmarks repeat. For growth, the doubling time is how long it takes for the quantity to double; for decay, the half-life is how long to halve. Both are properties of the rate alone — they don't depend on the starting amount.
If $P(t) = P_0 e^{kt}$ with $k > 0$, then doubling requires $e^{kt_d} = 2$, so $t_d = \dfrac{\ln 2}{k}$. The mirror identity for half-life with $k < 0$ is $t_{1/2} = \dfrac{\ln 2}{|k|}$.
3. The natural base $e$
Of all the possible bases, one stands apart. The number
$$ e = \lim_{n \to \infty}\!\left(1 + \frac{1}{n}\right)^{\!n} \approx 2.71828\ldots $$is the natural base of exponentials. It first appeared in finance — Jacob Bernoulli (1683) was studying compound interest and noticed that as the compounding frequency goes to infinity, the limit was an irrational number that wouldn't go away. Later Euler (1736) named it, studied it systematically, and proved it irrational.
Why "natural"?
The function $e^x$ has a property no other exponential does: it is its own rate of change. At every point on the curve, the slope equals the height. Any other base $a^x$ has slope $(\ln a) \cdot a^x$ — itself, but with an awkward multiplier in front. Only $e^x$ has that multiplier equal to $1$.
That's why every "rate is proportional to current size" problem — populations, decay, cooling, continuous interest, capacitor discharge — collapses onto $e^{kt}$ once you do the calculus. The $e$ isn't a stylistic choice; it falls out of the equation.
You'll see $e$ defined three ways, all of which give the same number. (i) The limit $\lim_{n \to \infty}(1 + 1/n)^n$ — financial origin. (ii) The infinite sum $\sum_{n=0}^{\infty} \tfrac{1}{n!} = 1 + 1 + \tfrac{1}{2} + \tfrac{1}{6} + \cdots$ — Taylor series. (iii) The unique number with $\int_1^e \tfrac{1}{t}\,dt = 1$ — area under $1/t$. Pick whichever your context prefers; calculus stitches them together.
Continuous compounding
Bernoulli's original setup is still the cleanest motivation. Suppose you invest $P$ at annual rate $r$, compounded $n$ times per year for $t$ years:
$$ A = P\!\left(1 + \frac{r}{n}\right)^{nt}. $$Crank $n$ up — quarterly, monthly, daily, every second — and the limit is:
$$ \lim_{n \to \infty} P\!\left(1 + \frac{r}{n}\right)^{nt} = P e^{rt}. $$That's continuous compounding. \$1000 at 6% for 5 years grows to $1000 \cdot e^{0.3} \approx \$1349.86$ — only about three dollars more than daily compounding gives you. The cap is $e^{rt}$.
4. Logarithm: the inverse
Exponentials ask "if I start at 1 and multiply by $b$ $x$ times, where do I end up?" Logarithms ask the reverse: "I ended up at $y$ — how many times did I multiply by $b$?" That's all a logarithm is.
$\log_b x = y$ means exactly $b^y = x$. The base $b > 0$ with $b \neq 1$; the argument $x > 0$.
Two pieces of notation worth memorizing once and never mixing up:
- Natural log: $\ln x = \log_e x$. Used everywhere in calculus, physics, and statistics.
- Common log: $\log x = \log_{10} x$ in engineering and chemistry contexts. In pure math and many physics texts, $\log$ alone often means $\ln$. Always check the field's convention.
- Binary log: $\log_2 x$. Native to computer science and information theory.
Two identities that do all the heavy lifting
Because $\log_b$ and $b^{(\cdot)}$ are inverses, applying one undoes the other:
$$ \log_b(b^x) = x \quad\text{for every real } x, \qquad b^{\log_b x} = x \quad\text{for every } x > 0. $$You will reach for these constantly when solving equations. They're how you "peel" a variable out of an exponent or out from inside a log.
Reading off easy values
| Expression | Equals | Because |
|---|---|---|
| $\log_2 8$ | $3$ | $2^3 = 8$ |
| $\log_{10} 1000$ | $3$ | $10^3 = 1000$ |
| $\ln e^5$ | $5$ | inverse identity |
| $\log_b 1$ | $0$ | $b^0 = 1$ for every $b$ |
| $\log_b b$ | $1$ | $b^1 = b$ |
5. The log laws
Three identities convert products, quotients, and powers inside a log into sums, differences, and coefficients outside. They're the same three laws of exponents in disguise — what's multiplicative on one side is additive on the other.
| Law | Identity | Comes from |
|---|---|---|
| Product | $\log_b(xy) = \log_b x + \log_b y$ | $b^{p+q} = b^p \cdot b^q$ |
| Quotient | $\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y$ | $b^{p-q} = b^p / b^q$ |
| Power | $\log_b(x^n) = n \log_b x$ | $(b^p)^n = b^{np}$ |
Two everyday uses:
- Expand: turn $\log\!\left(\frac{x^2 y}{z^3}\right)$ into $2\log x + \log y - 3 \log z$. Useful when you want to differentiate or simplify.
- Combine: turn $2\log x + 3\log y$ into $\log(x^2 y^3)$. Useful when solving an equation — collapse everything into a single log first.
$\log(x + y)$ does not equal $\log x + \log y$. The product law fires for $\log(xy)$, not $\log(x+y)$. This is the single most common error in early use of logs — and it's why the laws are stated for products, not sums.
Why logs were invented
Before electronic calculators, multiplying two seven-digit numbers was a half-hour ordeal. Napier (1614) noticed that if you tabulate $\log x$ for many $x$, then "multiply $a \cdot b$" becomes "look up $\log a$, look up $\log b$, add them, look up the antilog." Addition is easy. Multiplication is hard. The log table — and later the slide rule — used this identity to dominate engineering computation for 350 years.
6. Change of base
Your calculator has $\ln$ and $\log_{10}$. What if you need $\log_7 200$? The change-of-base formula bridges any two bases:
$$ \log_b x = \frac{\log_c x}{\log_c b} \qquad \text{for any valid base } c. $$In practice this means: pick whichever base your calculator supports — usually $e$ or $10$ — and divide.
$$ \log_7 200 = \frac{\ln 200}{\ln 7} \approx \frac{5.298}{1.946} \approx 2.723. $$The denominator carries the base. Mixing up which goes on top and which on the bottom is a frequent slip — remember that $\log_b b = 1$, so the base appears in the denominator (where it equals $\log_c b$).
Let $y = \log_b x$, so $b^y = x$. Take $\log_c$ of both sides: $\log_c(b^y) = \log_c x$, then by the power law $y \log_c b = \log_c x$, so $y = \log_c x / \log_c b$. One application of the power law is the whole proof.