Logarithm
Logarithms, laws, graphs, and logarithmic scales.
Number
Counts
About this topic
Logarithm Definition The exponent needed to produce a given number.
A logarithm answers the question: 'To what power must we raise the base to get this number?' It is the inverse operation of exponentiation.
Key formulas & rules
- •log_b(x) = y means bʸ = x
- •Base b must be > 0 and b ≠ 1
- •Domain: x > 0 only (no log of negatives)
- •log_b(1) = 0 for any valid base
- •log_b(b) = 1 (base to itself)
- •log_b(0) is undefined (−∞ limit)
Examples
| Input | Output | Note |
|---|---|---|
| log₂(8) | 3 | 2³ = 8 |
| log₁₀(100) | 2 | 10² = 100 |
| log₅(1) | 0 | Anything to power 0 = 1 |
| log₃(27) | 3 | 3³ = 27 |
Real-world uses
Logarithm Laws Rules that turn multiplicative problems into additive ones.
Logarithm laws convert multiplication into addition, division into subtraction, and powers into multiplication. These are the fundamental algebraic tools for working with logs.
Key formulas & rules
- •Product: log_b(xy) = log_b(x) + log_b(y)
- •Quotient: log_b(x/y) = log_b(x) − log_b(y)
- •Power: log_b(xⁿ) = n · log_b(x)
- •Root: log_b(ⁿ√x) = log_b(x) / n
- •Change of base: log_b(x) = logₖ(x) / logₖ(b)
- •Identity: b^(log_b(x)) = x
Examples
| Input | Output | Note |
|---|---|---|
| log(2·5) | log(2) + log(5) | Product rule |
| log(100/4) | log(100) − log(4) | Quotient rule |
| log(2⁵) | 5·log(2) | Power rule |
| log₂(100) | log(100)/log(2) | Change of base |
Real-world uses
Natural Logarithm (ln) Logarithm with base e ≈ 2.71828.
The natural log uses Euler's number e as its base. It appears everywhere in calculus, physics, and finance because it has the simplest derivative (d/dx ln(x) = 1/x).
Key formulas & rules
- •ln(x) = log_e(x) where e ≈ 2.71828
- •ln(e) = 1
- •ln(1) = 0
- •Inverse of eˣ: ln(eˣ) = x
- •Derivative: d/dx ln(x) = 1/x
- •Integral: ∫ (1/x) dx = ln|x| + C
Examples
| Input | Output | Note |
|---|---|---|
| ln(e) | 1 | Base to itself |
| ln(1) | 0 | Identity |
| ln(e³) | 3 | Inverse of exp |
| ln(10) | 2.3026 | Common constant |
Real-world uses
Common Logarithm (base 10) Logarithm with base 10 — used for scientific notation.
The common log uses base 10 and is written as log(x). It measures the number of digits in a number (roughly) and is the basis for the decibel and pH scales.
Key formulas & rules
- •log(x) = log₁₀(x)
- •log(10) = 1, log(100) = 2, log(1000) = 3
- •log(1) = 0
- •Approximates the digit-count minus 1
- •Basis of pH scale: pH = −log[H⁺]
- •Basis of decibel: dB = 10·log(P₁/P₂)
Examples
| Input | Output | Note |
|---|---|---|
| log(100) | 2 | 3 digits → 2 |
| log(1000) | 3 | 4 digits → 3 |
| log(0.01) | −2 | Small numbers → negative |
| log(7) | 0.8451 | Non-power of 10 |
Real-world uses
Logarithmic Scales Compress huge ranges into small ranges.
A logarithmic scale uses the log of a quantity instead of the quantity itself. This is essential when values span many orders of magnitude — like earthquakes, sound, or star brightness.
Key formulas & rules
- •Equal distances = equal ratios (not differences)
- •Common in scientific visualization
- •Compresses large ranges dramatically
- •Richter: each unit = 10× more energy
- •Decibel: +10 dB = 10× more power
- •pH: −1 = 10× more acidic
Examples
| Input | Output | Note |
|---|---|---|
| Richter 5 vs 7 | 100× more energy | 2 units = 10² |
| pH 3 vs 5 | 100× more acidic | Lower = more acidic |
| 60 dB vs 80 dB | 100× more power | 20 dB difference |
| Magnitude 6 vs 8 | 1000× | 10^1.5(2) |
Real-world uses
Real-World Applications Where logarithms solve real problems.
Logarithms appear in every field that deals with multiplicative processes, exponential growth, or extremely large/small numbers. They are one of the most practically useful mathematical tools.
Key formulas & rules
- •Finance: solve for time in compound interest
- •Biology: radioactive decay half-life
- •Chemistry: pH and pKa calculations
- •Physics: entropy, sound, earthquake magnitude
- •Computer Science: algorithm complexity (log n)
- •Statistics: log-likelihood, log-odds
Examples
| Input | Output | Note |
|---|---|---|
| t = ln(2)/r | Doubling time | Exponential growth |
| pH = −log[H⁺] | Acidity measure | Chemistry |
| O(log n) | Binary search | Fast algorithms |
| Half-life | t = ln(2)/λ | Decay |
Real-world uses
Modes & formulas
Basic Log
The base must be positive and not equal to 1. The argument x must be positive, since no positive base raised to any real power gives a non-positive result.
Log Laws
These laws convert multiplication into addition and division into subtraction, which is why logarithms were historically used as a computational shortcut.
Graph
The logarithm is defined only for x > 0. The graph grows without bound but very slowly. A base greater than 1 gives an increasing curve; a base between 0 and 1 gives a decreasing curve.
Log Scales
Logarithmic scales compress huge ranges into small ranges. A difference of 10 dB is a 10× power ratio, and a difference of 1 pH unit is a 10× change in hydrogen-ion concentration.