Logarithms — Mathematics Knowledge System

Overview

The logarithm of a number is the exponent to which a fixed base must be raised to produce that number. If x = bʸ, then y is the logarithm of x to base b, written log_b x = y. For example, log₁₀ 1000 = 3 because 10³ = 1000. The logarithm is the inverse of exponentiation, and it turns multiplication into addition — the property that made it, for three centuries, the backbone of hand computation, and today the natural language of exponential growth, decibels, pH, entropy, and algorithmic complexity.

Intuition

A logarithm answers one question: 'what power do I raise the base to, to get this number?' Ask 'log₂ 8?' — you're asking '2 to what power is 8?' Answer: 3. Because each step up in a logarithm multiplies the input by the base, logarithms compress huge multiplicative ranges into small additive ones: going from 100 to 1000 to 10000 (×10 each time) is just 2 → 3 → 4 in log₁₀. That is why doubling and order-of-magnitude thinking both live naturally on a log scale.

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Formal Definition

Definition

The logarithm base b (with b > 0, b ≠ 1) is defined as the inverse of exponentiation: log_b x is the unique exponent y satisfying bʸ = x, for x > 0.

\log_b x = y \iff b^y = x

Logarithm as the inverse of exponentiation (b > 0, b ≠ 1, x > 0)

Definition

b^{\log_b x} = x

Exponentiation undoes the logarithm

Notation

Notation	Meaning
$\log_b x$	Logarithm of x to base b
$\ln x$	Natural logarithm — base e (e ≈ 2.71828)
$\log x$	Common logarithm — base 10 (in engineering); base e or 2 in some fields
$\lg x$	Binary logarithm — base 2 (common in computer science)

Derivation

The product rule follows directly from the corresponding law of exponents. Let m = log_b x and n = log_b y, so bᵐ = x and bⁿ = y:

xy = b^m \cdot b^n

Substitute the exponential forms

xy = b^{m+n}

Law of exponents: bᵐ·bⁿ = bᵐ⁺ⁿ

\log_b(xy) = m + n

Take log_b of both sides (log and b^ are inverse)

\log_b(xy) = \log_b x + \log_b y

Substitute back m and n

Properties

Product rule

\log_b(xy) = \log_b x + \log_b y

Quotient rule

\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y

Power rule

\log_b(x^n) = n\log_b x

Log of base

\log_b b = 1

Log of one

\log_b 1 = 0

Change of base

\log_b x = \frac{\log_c x}{\log_c b}

Applications

The binary logarithm log₂ n counts how many times a size-n problem can be halved — the running time of binary search and the height of a balanced tree.

Worked Examples

Ask: 2 to what power is 32? Since 2⁵ = 32, the exponent is 5.
$2^5 = 32 \Rightarrow \log_2 32 = 5$

Answer: log₂ 32 = 5

Practice Problems

Difficulty 2/10

Evaluate log₁₀ 10000.

Difficulty 4/10

Write log_b(x/y²) as a sum/difference of logarithms.

Difficulty 3/10

What is log_b 1 for any valid base b?

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Common Mistakes

Common Mistake

Thinking log(x + y) equals log x + log y.

The product rule is about a PRODUCT inside the log: log(xy) = log x + log y. There is no simple rule for log(x + y) — it does not split.

Common Mistake

Writing log(x/y) = log x / log y.

The quotient rule gives a DIFFERENCE, not a quotient: log(x/y) = log x − log y. A ratio of logs (log x / log y) is instead the change-of-base expression for log_y x.

Quiz

Flashcards

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Historical Background

John Napier published the first table of logarithms in 1614 to speed up the multiplication of large numbers in astronomy and navigation, replacing multiplication with addition. Henry Briggs reworked Napier's idea into base-10 (common) logarithms in the 1620s. Leonhard Euler, in the 18th century, tied logarithms firmly to the exponential function and the constant e, giving the natural logarithm its modern footing.

1614
Napier publishes Mirifici Logarithmorum Canonis Descriptio, the first logarithm tables
John Napier
1624
Briggs publishes base-10 logarithms to 14 decimal places
Henry Briggs
1748
Euler formalizes the logarithm as the inverse of the exponential in Introductio in analysin infinitorum
Leonhard Euler

Summary

A logarithm log_b x is the exponent y such that bʸ = x — the inverse of exponentiation.
Product → sum: log_b(xy) = log_b x + log_b y; quotient → difference; power → multiplication.
Special values: log_b 1 = 0 and log_b b = 1.
Change of base log_b x = log_c x / log_c b lets any calculator's ln or log₁₀ compute any base.
Logarithms turn multiplicative, exponential-scale phenomena (dB, pH, complexity) into additive, linear ones.

References

BookStewart, J. Precalculus: Mathematics for Calculus, 7th ed. Ch. 4.
WebsiteWikipedia — Logarithm