Logarithm
A logarithm is the inverse operation of exponentiation: log_b(x) = y means b^y = x, answering the question "to what power must b be raised to produce x?" Logarithms transform multiplicative relationships into additive ones, making calculations with very large or small numbers tractable. They are central to information theory, signal processing, and scientific measurement scales such as pH, decibels, and the Richter scale.
Key Formula
log_b(x) = y ⟺ b^y = x, where b > 0, b ≠ 1, x > 0
LaTeX: \log_b(x) = y \iff b^y = x, \quad b > 0,\; b \neq 1,\; x > 0
| Symbol | Meaning | Unit |
|---|---|---|
| b | Base of the logarithm | dimensionless |
| x | Argument (input value) | dimensionless |
| y | Logarithmic value (exponent) | dimensionless |
Worked Example
Problem
Evaluate log₂(64) and log₁₀(0.001).
Solution
Part 1: log₂(64) = y means 2^y = 64. Since 2⁶ = 64, y = 6. Part 2: log₁₀(0.001) = y means 10^y = 0.001 = 10⁻³. So y = −3.
Answer
log₂(64) = 6; log₁₀(0.001) = −3.
Logarithm Laws
| Law | Formula | Example |
|---|---|---|
| Product rule | logₐ(xy) = logₐx + logₐy | log₁₀(100·10) = 2+1 = 3 |
| Quotient rule | logₐ(x/y) = logₐx − logₐy | log₂(8/2) = 3−1 = 2 |
| Power rule | logₐ(xⁿ) = n logₐx | log₁₀(10³) = 3 |
| Change of base | logₐx = log x / log a | log₂10 = 1/log10 2 ≈ 3.32 |
| Identity | logₐ(a) = 1 | log₅5 = 1 |
Interactive Tools
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Related Terms
Exponent
An exponent (also called a power or index) indicates how many times a base number is multiplied by itself. Written as aⁿ, where a is the base and n is the exponent, it represents repeated multiplication in a compact form. Exponents are essential in scientific notation, polynomial expressions, and exponential growth models.
Exponential Function
An exponential function is a mathematical function of the form f(x) = aˣ, where the variable x appears as the exponent and the base a is a positive constant not equal to 1. These functions exhibit rapid growth (when a > 1) or decay (when 0 < a < 1) and are fundamental models for population growth, compound interest, radioactive decay, and many natural phenomena.
Logarithmic Function
A logarithmic function is the inverse of an exponential function, defined as f(x) = log_b(x) for a positive base b ≠ 1 and positive input x. Its graph is a slowly increasing (or decreasing) curve that passes through (1, 0) and grows without bound, yet extremely slowly. Logarithmic functions model phenomena where large ranges of values need to be compressed, such as sound intensity, earthquake magnitude, and chemical acidity.
Coined by Scottish mathematician John Napier in 1614 from Greek "logos" (ratio) and "arithmos" (number), meaning "ratio number." Napier introduced logarithms to simplify astronomical calculations.