Logarithmic Functions

The logarithm is the inverse of exponentiation: $lo g_{b} x$ answers "to what power must $b$ be raised to get $x$ ?" Logarithmic functions turn multiplication into addition, taming exponential growth and powering scales from decibels to pH. Used across science, finance, and computing; formalized by John Napier (1614) and reframed as an inverse function by Euler in the 1700s.

A logarithmic function is a function of the form $f (x) = lo g_{b} x$ , where the base $b$ is a positive real number not equal to 1, and the argument $x$ is a positive real number. It is precisely the inverse of the exponential function $g (x) = b^{x}$ : the logarithm undoes exponentiation. Logarithms matter because they convert the hard operation of multiplication into the easy operation of addition, a property that drove three centuries of scientific computation and underlies the structure of countless natural and engineered systems. The surprising fact: although logarithms feel like an advanced topic, they were invented before the modern concept of exponents was fully developed — Napier built them to multiply numbers, not to invert powers.

The Core Insight

A logarithm is an exponent in disguise. The statement $lo g_{b} x = y$ is just another way of writing $b^{y} = x$ . Everything about logarithms flows from this single equivalence. Because exponents add when you multiply powers ( $b^{m} \cdot b^{n} = b^{m + n}$ ), logarithms convert products into sums. This is the structural reason logarithms exist and the reason they are useful.

Formal Statement

Logarithmic Function

For a fixed base $b$ with $b > 0$ and $b \neq = 1$ , the logarithm base $b$ is defined by the equivalence:

y = lo g_{b} x ⟺ b^{y} = x (x > 0)

⟳This is a definition by inverse — read it as "the logarithm is the exponent." Both directions are the same statement.

Here $b$ is the base, $x$ is the argument (or antilogarithm), and $y$ is the logarithm (the resulting exponent). The domain is $x \in (0, \infty)$ and the range is $y \in (- \infty, \infty)$ .

Three special bases have dedicated notation:

Common logarithm $lo g_{10} x$ , written $lo g x$ (engineering, scales).
Natural logarithm $lo g_{e} x$ , written $ln x$ , where $e \approx 2.71828$ (calculus, growth).
Binary logarithm $lo g_{2} x$ , written $l g x$ (computer science).

Laws of Logarithms

For $b > 0$ , $b \neq = 1$ , and positive arguments $M, N$ , and any real $p$ :

lo g_{b} (M N) = lo g_{b} M + lo g_{b} N

⟳Let

lo g_{b} M = m

lo g_{b} N = n

, so

M = b^{m}

N = b^{n}

; then

M N = b^{m + n}

lo g_{b} (\frac{M}{N}) = lo g_{b} M - lo g_{b} N

⟳Same substitution;

M / N = b^{m - n}

, so the logarithm is

m - n

lo g_{b} (M^{p}) = p lo g_{b} M

⟳

M^{p} = (b^{m})^{p} = b^{m p}

, so the exponent is

m p = p lo g_{b} M

lo g_{b} x = \frac{lo g _{a} x}{lo g _{a} b} (change of base)

⟳From

x = b^{l o g_{b} x}

, take

lo g_{a}

of both sides and apply law [4].

Identities and Special Values

The definitional equivalence forces several identities directly:

lo g_{b} 1 = 0, lo g_{b} b = 1, lo g_{b} (b^{x}) = x, b^{l o g_{b} x} = x

⟳Set

b^{0} = 1

b^{1} = b

; the last two are the inverse-composition statements

f^{- 1} (f) = id

and

f (f^{- 1}) = id

Why It Holds / Derivation

The Product Law from the Exponent Law

We prove equation [2], $lo g_{b} (M N) = lo g_{b} M + lo g_{b} N$ . Every other law follows by the same technique.

Name the logarithms. Let $m = lo g_{b} M$ and $n = lo g_{b} N$ . By the definition [1], this means $M = b^{m}$ and $N = b^{n}$ .

Multiply the arguments. Form the product:

M N = b^{m} \cdot b^{n} .

Apply the exponent addition law. Powers of the same base multiply by adding exponents, so:

M N = b^{m + n} .

Convert back using the definition. The equation $M N = b^{m + n}$ says, by [1], that $lo g_{b} (M N) = m + n$ .

Substitute the names back. Replacing $m$ and $n$ :

lo g_{b} (M N) = lo g_{b} M + lo g_{b} N . ■

The entire content of the proof is that logarithms inherit the additive structure of exponents. The quotient law [3] uses $b^{m} / b^{n} = b^{m - n}$ ; the power law [4] uses $(b^{m})^{p} = b^{m p}$ .

John Napier | 1550–1617

A Scottish landowner, theologian, and mathematician, Napier spent twenty years computing his tables of logarithms, published in 1614 as Mirifici Logarithmorum Canonis Descriptio. His motivation was purely practical: astronomers and navigators drowned in multiplying long strings of trigonometric values. Napier's insight was to relate two points moving along lines — one at constant speed, one at speed proportional to its remaining distance — producing a correspondence that turned multiplication into addition. His "logarithms" (from Greek logos, ratio, + arithmos, number) were not yet tied to a clean base, but they launched the modern computational era. Henry Briggs later collaborated with him to rebase the tables on 10.

Building Intuition

Think of a logarithm as counting digits, or counting orders of magnitude. The common log $lo g_{10}$ of a number tells you roughly how many digits it has: $lo g_{10} (1000) = 3$ , and $1000$ has three zeros. More precisely, $lo g_{b} x$ answers "how many times must I multiply by $b$ to reach $x$ ?" — and it allows fractional answers for the in-between values.

The second intuition is compression of scale. Multiplying by 10 only adds 1 to the logarithm. So a logarithmic axis squeezes a vast range — from 1 to 1,000,000 — into a tidy span from 0 to 6. This is why "log scales" appear whenever quantities span enormous ranges (earthquakes, sound, acidity, star brightness).

The third intuition is growth that slows. As $x$ increases, $lo g_{b} x$ keeps rising but ever more sluggishly, because each new unit of logarithm requires multiplying $x$ by another whole factor of $b$ . The curve flattens but never stops, and never turns back down.

The natural logarithm $y=\ln x$, its inverse $y=e^x$, and the line $y=x$

Worked Examples

Solving an Exponential Equation with Logs

Given: A bacterial colony doubles every 3 hours, starting at 500 cells. After how many hours does it reach 32,000 cells? Model:

N (t) = 500 \cdot 2^{t /3}

. Set

N (t) = 32000

. Steps:

Divide: $\frac{32000}{500} = 64$ , so $2^{t /3} = 64$ .
Recognize $64 = 2^{6}$ , so $2^{t /3} = 2^{6}$ gives $\frac{t}{3} = 6$ .
(General method, if the factor weren't a clean power) take logs:

\frac{t}{3} = lo g_{2} 64 = \frac{ln 64}{ln 2} = \frac{4.1589}{0.6931} = 6.

Solve: $t = 18$ .

t = 18 hours

Earthquake Magnitudes — A Log-Scale Comparison

Given: The Richter magnitude is

M = lo g_{10} (A / A_{0})

, where

A

is the measured amplitude and

A_{0}

a reference. How many times stronger (in amplitude) is a magnitude 7.2 quake than a magnitude 5.4 quake? Steps:

Write each: $M_{1} = lo g_{10} (A_{1} / A_{0}) = 7.2$ and $M_{2} = lo g_{10} (A_{2} / A_{0}) = 5.4$ .
Subtract using the quotient law [3]:

M_{1} - M_{2} = lo g_{10} (\frac{A _{1}}{A _{2}}) = 7.2 - 5.4 = 1.8.

Undo the log: $\frac{A _{1}}{A _{2}} = 1 0^{1.8}$ .
Evaluate: $1 0^{1.8} = 1 0^{1} \cdot 1 0^{0.8} \approx 10 \times 6.31 = 63.1$ .

about 63 times larger amplitude

Common Misconceptions

The logarithm of a sum is NOT the sum of logarithms. A very common error is to write

lo g_{b} (M + N) = lo g_{b} M + lo g_{b} N

. This is false. The product law [2] applies only to products:

lo g_{b} (M N) = lo g_{b} M + lo g_{b} N

. There is no simplification for

lo g_{b} (M + N)

Other frequent mistakes:

$\frac{lo g _{b} M}{lo g _{b} N} \neq = lo g_{b} (\frac{M}{N})$ . The quotient law gives $lo g_{b} M - lo g_{b} N$ , a subtraction, not a division of logs.
$(lo g_{b} M)^{p} \neq = p lo g_{b} M$ . The power law [4] requires the exponent to be inside the argument: $lo g_{b} (M^{p}) = p lo g_{b} M$ .
$lo g_{b} 0$ and $lo g_{b} (negative)$ are undefined in the reals — no real exponent of a positive base yields zero or a negative number.

Test the false "sum rule" with numbers. Take $b = 10$ , $M = 100$ , $N = 100$ :

lo g_{10} (M + N) = lo g_{10} (200) \approx 2.301,

but

lo g_{10} M + lo g_{10} N = 2 + 2 = 4.

Since

2.301 \neq = 4

, the sum rule is conclusively wrong. (The correct product law:

lo g_{10} (100 \cdot 100) = lo g_{10} (10000) = 4 = 2 + 2

. ✓)

Connections

Logarithmic functions sit at the center of a web of related ideas. They are the inverse of Exponential Functions, so every property of one mirrors a property of the other — reflection across $y = x$ . The base $e$ ties logarithms to The Number e and makes $ln x$ the natural choice in Calculus, where $\frac{d}{d x} ln x = \frac{1}{x}$ and $\int \frac{1}{x} d x = ln ∣ x ∣$ . The change-of-base formula [5] connects logarithms in any base, and the laws [2]–[4] are a concrete instance of a Group Isomorphism between multiplication of positive reals and addition of all reals. Logarithms also underlie Inverse Functions generally and feed directly into the study of Exponential Growth and Decay.

Real-World Applications

Decibels (sound): Loudness $L = 10 lo g_{10} (I / I_{0})$ dB compresses the enormous range of audible sound intensities into a manageable 0–120 scale.
pH (chemistry): Acidity $pH = - lo g_{10} [H^{+}]$ turns hydrogen-ion concentrations spanning many orders of magnitude into the familiar 0–14 scale.
Richter / moment magnitude (seismology): Earthquake energy spans factors of billions; logs make magnitudes comparable (see worked example).
Information theory (computing): The number of bits to encode $N$ equally likely states is $lo g_{2} N$ ; Shannon entropy is a sum of $- p lo g_{2} p$ terms.
Finance: Continuously compounded returns use $ln (P_{1} / P_{0})$ , which add cleanly across periods, unlike percentage returns.
Astronomy (stellar magnitudes): Apparent brightness is logarithmic; a difference of 5 magnitudes equals a factor of 100 in flux.
Algorithm analysis (CS): Binary search and balanced trees run in $O (lo g n)$ time — the log measures how many halvings reach a single element.

Historical Development

1614 — John Napier publishes Mirifici Logarithmorum Canonis Descriptio, introducing logarithms to simplify trigonometric multiplication for astronomers and navigators.
1617 — Henry Briggs visits Napier; together they conceive base-10 ("common") logarithms with $lo g 1 = 0$ and $lo g 10 = 1$ .
1620 — Joost Bürgi independently publishes his own progression-based tables, developed before Napier's but printed later.
1624 — Briggs publishes Arithmetica Logarithmica, tables of common logs to 14 decimal places for 1–20,000 and 90,000–100,000.
1622 — William Oughtred invents the slide rule, a physical embodiment of the logarithm that would dominate calculation for 350 years.
1647–1668 — Gregory of Saint-Vincent and Nicholas Mercator link the area under the hyperbola $1/ x$ to the logarithm, birthing the natural log and its series $ln (1 + x) = x - \frac{x ^{2}}{2} + \dots$ .
1748 — Leonhard Euler, in Introductio in analysin infinitorum, defines logarithms as inverses of exponentials, cements the constant $e$ , and gives the modern viewpoint.
1850s onward — Logarithm tables and slide rules become standard in engineering and science education worldwide.
1970s — Electronic calculators render printed tables and slide rules obsolete, though the underlying functions remain everywhere in mathematics.

Subtleties & Limits

The clean real-valued logarithm hides genuine subtleties once the domain widens. Extending $lo g$ to negative and complex numbers forces a choice: $lo g z = ln ∣ z ∣ + i ar g z$ , but $ar g z$ is determined only up to multiples of $2 π$ , so the complex logarithm is multi-valued. Selecting a "principal branch" introduces a discontinuity (a branch cut), and there is no single global continuous logarithm on the punctured plane — a fact tied to deep ideas in Complex Analysis and topology. Even on the reals, the question "what is the simplest base?" has a non-obvious answer: $e$ wins because it makes the derivative cleanest, yet $e$ itself is transcendental and was not isolated until long after logarithms were in use. Finally, whether certain logarithms of integers are rational or transcendental (e.g., is $lo g_{10} 2$ irrational? — yes, provably) opens into number theory, where many related values resist full classification.