Logarithmic Functions

⚡ Key Takeaways

The logarithm is the inverse of exponentiation: $\log_b x$ 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.

💡 The Core Insight

A logarithm is an exponent in disguise. The statement $\log_b x = y$ 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.

DefinitionLogarithmic Function

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

1

\quad y = \log_b x \iff b^y = x \qquad (x > 0)

Derivation hint: 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 $\log_{10} x$ , written $\log x$ (engineering, scales).
Natural logarithm $\log_e x$ , written $\ln x$ , where $e \approx 2.71828$ (calculus, growth).
Binary logarithm $\log_2 x$ , written $\lg x$ (computer science).

TheoremLaws of Logarithms

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

2

\quad \log_b(MN) = \log_b M + \log_b N

Derivation hint: Let $\log_b M = m$ , $\log_b N = n$ , so $M = b^m$ , $N = b^n$ ; then $MN = b^{m+n}$ .

3

\quad \log_b\!\left(\frac{M}{N}\right) = \log_b M - \log_b N

Derivation hint: Same substitution; $M/N = b^{m-n}$ , so the logarithm is $m - n$ .

4

\quad \log_b(M^p) = p\,\log_b M

Derivation hint: $M^p = (b^m)^p = b^{mp}$ , so the exponent is $mp = p\log_b M$ .

5

\quad \log_b x = \frac{\log_a x}{\log_a b} \qquad \text{(change of base)}

Derivation hint: From $x = b^{\log_b x}$ , take $\log_a$ of both sides and apply law [4].

TheoremIdentities and Special Values

The definitional equivalence forces several identities directly:

6

\quad \log_b 1 = 0, \qquad \log_b b = 1, \qquad \log_b(b^x) = x, \qquad b^{\log_b x} = x

Derivation hint: Set $b^0 = 1$ , $b^1 = b$ ; the last two are the inverse-composition statements $f^{-1}(f) = \mathrm{id}$ and $f(f^{-1}) = \mathrm{id}$ .

ProofThe Product Law from the Exponent Law

We prove equation [2], $\log_b(MN) = \log_b M + \log_b N$ . Every other law follows by the same technique. 1. Name the logarithms. Let $m = \log_b M$ and $n = \log_b N$ . By the definition [1], this means $M = b^m$ and $N = b^n$ . 2. Multiply the arguments. Form the product:

MN = b^m \cdot b^n.

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

MN = b^{m+n}.

4. Convert back using the definition. The equation $MN = b^{m+n}$ says, by [1], that $\log_b(MN) = m + n$ . 5. Substitute the names back. Replacing $m$ and $n$ :

\log_b(MN) = \log_b M + \log_b N. \qquad \blacksquare

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 $(b^m)^p = b^{mp}$ .

∎

JN

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.

🧠 Intuition

Think of a logarithm as counting digits, or counting orders of magnitude. The common log $\log_{10}$ of a number tells you roughly how many digits it has: $\log_{10}(1000) = 3$ , and $1000$ has three zeros. More precisely, $\log_b x$ 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, $\log_b x$ 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.

Figure

The natural logarithm

y=\ln x

, its inverse

y=e^x

, and the line

y=x

ExampleSolving 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: 1. Divide: $\dfrac{32000}{500} = 64$ , so $2^{t/3} = 64$ . 2. Recognize $64 = 2^6$ , so $2^{t/3} = 2^6$ gives $\dfrac{t}{3} = 6$ . 3. (General method, if the factor weren't a clean power) take logs:

\frac{t}{3} = \log_2 64 = \frac{\ln 64}{\ln 2} = \frac{4.1589}{0.6931} = 6.

4. Solve: $t = 18$ .

\boxed{t = 18 \text{ hours}}

ExampleEarthquake Magnitudes — A Log-Scale Comparison

Given: The Richter magnitude is $M = \log_{10}(A/A_0)$ , 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: 1. Write each: $M_1 = \log_{10}(A_1/A_0) = 7.2$ and $M_2 = \log_{10}(A_2/A_0) = 5.4$ . 2. Subtract using the quotient law [3]:

M_1 - M_2 = \log_{10}\!\left(\frac{A_1}{A_2}\right) = 7.2 - 5.4 = 1.8.

3. Undo the log: $\dfrac{A_1}{A_2} = 10^{1.8}$ . 4. Evaluate: $10^{1.8} = 10^{1}\cdot 10^{0.8} \approx 10 \times 6.31 = 63.1$ .

\boxed{\text{about } 63 \text{ times larger amplitude}}

⚠️ Common mistake

The logarithm of a sum is NOT the sum of logarithms. A very common error is to write $\log_b(M + N) = \log_b M + \log_b N$ . This is false. The product law [2] applies only to products: $\log_b(MN) = \log_b M + \log_b N$ . There is no simplification for $\log_b(M+N)$ . Other frequent mistakes:

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

Counterexample

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

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

but

\log_{10}M + \log_{10}N = 2 + 2 = 4.

Since $2.301 \neq 4$ , the sum rule is conclusively wrong. (The correct product law: $\log_{10}(100 \cdot 100) = \log_{10}(10000) = 4 = 2 + 2$ . ✓)

connection

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 $\frac{d}{dx}\ln x = \frac{1}{x}$ and $\int \frac{1}{x}\,dx = \ln|x|$ . 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.

🔬 Where this appears

⏳ History

- 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

\log 1 = 0

and

\log 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-\tfrac{x^2}{2}+\cdots

.

- 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.

open-problem

The clean real-valued logarithm hides genuine subtleties once the domain widens. Extending $\log$ to negative and complex numbers forces a choice: $\log z = \ln|z| + i\arg z$ , but $\arg z$ is determined only up to multiples of $2\pi$ , 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 $\log_{10} 2$ irrational? — yes, provably) opens into number theory, where many related values resist full classification.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also