What is a logarithm?

The logarithm is the inverse operation of exponentiation. If a^b = c, then log_a(c) = b. In words: the base-a logarithm of c is the exponent to which you must raise a to get c.

Example: 2³ = 8, so log₂(8) = 3.

Three most common bases

1. Common logarithm — base 10

log(x) or log₁₀(x): historical calculation tool, now used in acoustics (decibels), seismology (Richter), chemistry (pH), astronomy (magnitudes).

• log(1) = 0
• log(10) = 1
• log(100) = 2
• log(1000) = 3

2. Natural logarithm — base e

ln(x): where e ≈ 2.71828 is Euler's constant. Fundamental in calculus, probability, physics.

• ln(1) = 0
• ln(e) = 1
• ln(10) ≈ 2.3026

3. Binary logarithm — base 2

log₂(x): essential in computer science (algorithm complexity, Shannon information theory).

• log₂(1) = 0
• log₂(2) = 1
• log₂(8) = 3
• log₂(1024) = 10

Fundamental properties

Valid for any base (b > 0, b ≠ 1):

1. Product rule

log_b(x·y) = log_b(x) + log_b(y)

Transforms multiplication into addition. This property made logarithms revolutionary for astronomical and navigational calculations (Napier 1614).

2. Quotient rule

log_b(x/y) = log_b(x) − log_b(y)

3. Power rule

log_b(xⁿ) = n · log_b(x)

4. Change of base

log_b(x) = log_c(x) / log_c(b)

5. Notable identities

• log_b(1) = 0
• log_b(b) = 1
• log_b(bⁿ) = n
• b^log_b(x) = x

Why e exists

The number e ≈ 2.71828 is the natural base of logarithms because:

d/dx ln(x) = 1/x (simplest possible derivative)

e emerges naturally in:

• Continuous compound interest: lim(1 + 1/n)ⁿ = e (Bernoulli 1683)
• Growth/decay differential equations
• Normal distribution
• Quantum mechanics

History of logarithms

• ~1594-1614 — John Napier: develops logarithms, publishes Mirifici Logarithmorum Canonis Descriptio (1614). Revolutionizes astronomical calculations.
• 1620 — Edmund Gunter: introduces slide rule
• 1624 — Henry Briggs: publishes base-10 logarithm tables
• 1683 — Jacob Bernoulli: discovers e in compound interest problem
• 1748 — Leonhard Euler: introduces symbol e, proves irrationality
• 1873 — Charles Hermite: proves e transcendental
• 1970s — Electronic calculators: make slide rule obsolete

The slide rule

From 1620 to 1970, engineers used the slide rule: two logarithmic scales sliding against each other. Using log(xy) = log(x) + log(y), multiplication became physical addition of lengths.

Precision: 3-4 digits. Apollo 11 moon mission (1969) was calculated with slide rules.

Natural logarithmic scales

Earthquakes — Richter scale

Magnitude M = log₁₀(A/A₀). M6 earthquake is 10x stronger than M5, and 32x more energetic.

Sound — decibels (dB)

dB = 10·log₁₀(P/P₀). Conversation ~60 dB, rock concert ~120 dB. 60 dB difference = factor 1,000,000 in intensity!

Acidity — pH

pH = −log₁₀([H⁺]). pH 4 solution is 1,000x more acidic than pH 7.

Stars — magnitudes

Stellar magnitude scale is logarithmic. 5 magnitudes difference = 100x in brightness.

Music frequencies

We perceive notes logarithmically: doubling frequency = one octave (12 semitones).

Finance — log-returns

Quantitative finance prefers log-returns r = ln(P_t/P_t-1) because they're time-additive and symmetric.

Algorithm complexity

Logarithms describe efficient algorithm speed:

• O(log n): binary search (halving search space each step)
• O(n log n): efficient sorts (merge sort, quick sort average)
• O(log log n): advanced data structures (Van Emde Boas)

O(log n) on 1 billion elements requires only ~30 operations.

Practical examples

Example 1 — Digits of a number

How many digits in 2¹⁰⁰⁰? log₁₀(2¹⁰⁰⁰) = 1000 · 0.30103 = 301.03. So 2¹⁰⁰⁰ has 302 digits.

Example 2 — Half-life time

Substance decays 5% yearly. How long to 50% reduction? t = ln(0.5)/ln(0.95) ≈ 13.5 years.

Example 3 — Compound interest

$10,000 at 6% annually, how long to double? t = ln(2)/ln(1.06) ≈ 11.9 years. Confirms Rule of 72: 72/6 = 12.

Example 4 — Earthquake magnitude

Japan 2011 M9.1. How much more energetic than M5.0? 10^9.1−5 × 32^(9.1−5) ≈ 31,623 in amplitude, ~5.6 million in energy.

Example 5 — Shannon information

How many bits for 1000 equiprobable states? log₂(1000) ≈ 9.97. Need 10 bits (2^10 = 1024 ≥ 1000).

Logarithms in equations

Equation 3^x = 50 solves with logarithms:

x · log(3) = log(50) → x = log(50)/log(3) ≈ 3.561

Used for population growth, pharmacokinetics, radioactive decay, compound finance.

Logarithm integrals

∫ ln(x) dx = x·ln(x) − x + C

∫ (1/x) dx = ln|x| + C (fundamental!)

1/x has no elementary antiderivative except the logarithm. This is the true mathematical reason for ln's importance.

Complex logarithm

Extended to complex numbers via Euler's formula:

e^iθ = cos θ + i·sin θ

Surprising consequence: ln(−1) = iπ

Euler's identity: e^iπ + 1 = 0. Considered math's most beautiful formula (Feynman, Russell).

Antilogarithm

The antilogarithm is the inverse operation: if log_b(x) = y, then antilog = b^y = x. Simply the exponential function.

Excel / Sheets / Python functions

Excel / Google Sheets

• =LOG(x): log₁₀(x) in Excel, natural log in Sheets (caution!)
• =LOG10(x): always base 10 (both)
• =LN(x): natural logarithm
• =LOG(x, base): any base
• =EXP(x): e^x
• =10^x: antilog base 10

Python (math, numpy)

import math
math.log(100)          # 4.605 (ln)
math.log10(100)        # 2.0
math.log2(8)           # 3.0
math.log(100, 10)      # 2.0
math.exp(1)            # 2.718...

import numpy as np
np.log(np.array([1, 10, 100]))

Common mistakes

• log(0): undefined, tends to −∞
• log(negative): undefined in reals
• "log" ambiguity: in pure math often = ln; engineering = log₁₀; CS = log₂
• log(x+y) ≠ log(x) + log(y): common error! Only for multiplication
• Forgetting properties: log(x²) = 2·log(x), not log(x)²

FAQ

Why are logarithms useful?

Transform exponential growth into linear scales, multiplication into addition, powers into multiplication. Essential when variables span many orders of magnitude.

log or ln: which to use?

Context-dependent. ln in theoretical math/physics (simple derivatives). log₁₀ in engineering. log₂ in computer science.

How to read log-log plot?

Both axes logarithmic. A straight line in log-log represents power law y = ax^k.

Why is e called "Napier's number"?

Actually Napier didn't discover it: used a close base but wasn't aware of the constant. Euler named it e in 1748.

Updated July 2026