0.2 Logarithms and exponentials

The exponential is the function of constant relative growth: it multiplies by a fixed factor over each fixed step. Its inverse, the logarithm, turns multiplication into addition and ratios into differences. Together they are the natural language for any quantity that ranges over orders of magnitude — and the bookshelf is full of these. Sound pressures run from a $20\,\mu\text{Pa}$ threshold to pain six decades higher; audible frequencies span ten octaves; nerves fire at rates that vary multiplicatively. The decibel, the octave, and the log-frequency axis of every audiogram are all the same idea: measure ratios, and you are measuring with logarithms.

Exponential laws

For a fixed base $a > 0$ , the exponential $a^x$ obeys

a^{x+y} = a^x a^y,\qquad a^{x-y} = \frac{a^x}{a^y},\qquad (a^x)^y = a^{xy},\qquad a^0 = 1.

The first law is the whole character of the function: adding in the exponent multiplies the value. It is what forces $a^0 = 1$ (so that $a^{x+0} = a^x$ ) and what extends the definition from positive-integer powers — where $a^n$ is $a$ multiplied $n$ times — to zero, negative, and fractional exponents by insisting the law keep holding.

▶ From repeated multiplication to fractional exponents Derivation

For positive integers, $a^n$ means $a\cdot a\cdots a$ ( $n$ factors), and $a^{m+n} = a^m a^n$ is just counting factors. To extend to all real exponents we demand that law persist and read off what each new symbol must mean.

$a^0$ : from $a^{n+0} = a^n a^0$ we need $a^0 = 1$ .
$a^{-n}$ : from $a^{n}a^{-n} = a^{0} = 1$ we need $a^{-n} = 1/a^n$ .
$a^{1/q}$ : from $\bigl(a^{1/q}\bigr)^q = a^{1} = a$ we need $a^{1/q} = \sqrt[q]{a}$ , the $q$ -th root.
$a^{p/q}$ : then $a^{p/q} = \bigl(\sqrt[q]{a}\bigr)^p$ .

Each extension is forced, not chosen — the single requirement $a^{x+y}=a^x a^y$ pins down the function on the rationals, and continuity fills in the irrationals. ✓

The natural base $e$

Among all bases, one is singled out by calculus rather than convenience. The number

e = \lim_{n\to\infty}\Bigl(1 + \tfrac1n\Bigr)^n = 2.71828\dots

is the base for which the exponential is its own derivative:

\boxed{\;\frac{d}{dx}e^x = e^x.\;}

No other base has slope equal to height at every point; for a general base, $\frac{d}{dx}a^x = (\ln a)\,a^x$ carries an extra factor $\ln a$ , which is 1 exactly when $a = e$ . This is why $e^x$ — and not $2^x$ or $10^x$ — is the exponential that appears the instant a rate of change is proportional to an amount: radioactive decay, charging capacitors, cooling bodies, the decay of every transient on the bookshelf. The series and the equilibrium-linearisation that make $e^x$ central are developed in Foundations 1.3 — Taylor series; here it is enough to know that $e$ is the base in which the calculus is frictionless.

Logarithms as inverses

The logarithm base $a$ answers the question to what power must I raise $a$ to get $x$ :

y = \log_a x \iff a^y = x.

Because it undoes an exponential, each exponential law becomes a logarithm law — and crucially, the law that turns multiplication into addition:

\boxed{\;\log_a(xy) = \log_a x + \log_a y,\;}\qquad \log_a\!\Bigl(\frac{x}{y}\Bigr) = \log_a x - \log_a y,\qquad \log_a(x^p) = p\,\log_a x.

The natural logarithm $\ln x = \log_e x$ is the inverse of $e^x$ ; the common logarithm $\log_{10} x$ is the one in which “an order of magnitude” is one unit, and the one the decibel is built on.

▶ The product law is the exponential law read backwards Derivation

Let $u = \log_a x$ and $v = \log_a y$ , so by definition $a^u = x$ and $a^v = y$ . Multiply:

xy = a^u a^v = a^{u+v}

by the exponential addition law. Taking $\log_a$ of both ends — which extracts the exponent — gives $\log_a(xy) = u + v = \log_a x + \log_a y$ . The logarithm inherits its defining property directly from the exponential it inverts: the exponential adds exponents to multiply, so the logarithm splits products into sums. ✓

Change of base

A logarithm in one base is a logarithm in any other up to a constant factor:

\log_a x = \frac{\log_b x}{\log_b a}.

The practical consequences are that $\log_2$ , $\ln$ , and $\log_{10}$ differ only by overall scale — $\ln x = \ln(10)\,\log_{10}x \approx 2.303\,\log_{10}x$ — so “logarithmic” is a single shape regardless of base, and a log axis looks the same whether labelled in octaves ( $\log_2$ ), decades ( $\log_{10}$ ), or natural units.

Thinking in decades and decibels

A linear axis cannot show a quantity that spans six orders of magnitude: anything more than a factor of a hundred below the maximum is crushed into the origin. A logarithmic axis spends equal space on equal ratios, so each factor of ten — each decade — gets its own stretch.

x = 100 = 10^2.00

log₁₀ x = 2.00 ln x = 4.61 10·log₁₀ x = 20.0 dB 20·log₁₀ x = 40.0 dB

Drag the value and watch the two markers. On the linear axis the marker is pinned near zero until $x$ approaches the top decade; on the log axis it moves smoothly, because the slider controls $\log_{10}x$ directly and equal slider motion is equal ratio. This is the whole reason acoustics measures in decibels. A level in dB is a logarithm of a ratio, rescaled:

L = 10\log_{10}\!\frac{P}{P_0}\ \text{(power)},\qquad L = 20\log_{10}\!\frac{A}{A_0}\ \text{(amplitude)},

the factor of 20 for amplitude being the factor of 10 for power with the power-law $\log(A^2) = 2\log A$ folded in, since power goes as amplitude squared. A factor of ten in power is $10\,\text{dB}$ ; a doubling is about $3\,\text{dB}$ . The same instinct names musical pitch: an octave is a doubling of frequency, $\log_2$ of the frequency ratio, and the audiogram’s frequency axis is logarithmic for exactly this reason.

Exponential growth and decay

When a quantity changes at a rate proportional to its current size, it is exponential. Decay is the case the bookshelf meets most:

x(t) = x_0\, e^{-t/\tau},

with time constant $\tau$ the time to fall to $1/e \approx 37\%$ of the start. The half-life $t_{1/2} = \tau\ln 2 \approx 0.693\,\tau$ is the same fact in base 2. On a log axis an exponential decay is a straight line of slope $-1/\tau$ , which is how a ring-down or a reverberation tail is read off in practice. The differential equation behind this, $\dot x = -x/\tau$ , is the first one solved in Foundations 5.2 — First-order linear ODEs.

Check yourself

Two uncorrelated sources each produce $60\,\text{dB}$ SPL at a microphone. What is the combined level, and why is it not $120\,\text{dB}$ ?

Reveal answer

Decibels are logarithms, so they do not add — the underlying powers add. Two equal powers double the total, and $10\log_{10}2 \approx 3\,\text{dB}$ , so the combined level is about $63\,\text{dB}$ , not $120$ . Adding the dB values would multiply the powers, which is meaningless here. This $+3\,\text{dB}$ -per-doubling rule is the everyday face of $\log(2)\approx 0.3$ .

⏳ The history — Napier's bones-deep idea, Briggs's base ten, Euler's e

John Napier published the first table of logarithms in 1614 (Napier 1614), with the explicit aim of replacing the multiplication of large numbers — a daily burden in astronomy and navigation — by addition. His logarithms were defined kinematically, by comparing a point moving at constant speed with one moving at speed proportional to its distance remaining, and were not quite our natural logarithm, but the central property, products become sums, was there from the start. Henry Briggs visited Napier and together they recast the tables to base 10 (Briggs 1624), the form that dominated calculation for three centuries through printed tables and the slide rule. The constant $e$ and the natural logarithm came later, from the calculus: Euler named $e$ and established its central role in the Introductio (Euler 1748), where the exponential and the trigonometric functions are unified. The decibel is a direct descendant — Bell Labs’ logarithmic unit for ratios of power, named for Alexander Graham Bell.

What we use it for

Logarithms and exponentials run through every book on the shelf:

The decibel scale for sound level — the foundation of audiometry and the audiogram (Tools of Audiology Ch 2).
Exponential decay of every transient: damped oscillators, reverberation, the first-order ODE (Foundations 5.2).
Log-frequency axes and octaves — the tonotopic, roughly logarithmic frequency map of the cochlea (Hearing Ch 4).
The complex exponential $e^{i\theta}$ , the exponential continued to imaginary arguments, on which all phasor and Fourier methods rest (Foundations Ch 3).
Log-scaled perception more broadly — the near-logarithmic compression of intensity and the multiplicative structure of just-noticeable differences.

What’s next

This closes the pre-calculus chapter. With trigonometry and logarithms restored to working order, the single-variable calculus chapter can take them as given — differentiating sinusoids in radians, and treating $e^x$ as the function equal to its own derivative — and build the derivative, the integral, and the Taylor series on top.