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.
Key Formula
f(x) = log_b(x) = ln(x) / ln(b), for x > 0, b > 0, b ≠ 1
LaTeX: f(x) = \log_b(x) = \frac{\ln x}{\ln b}, \quad x > 0,\; b > 0,\; b \neq 1
| Symbol | Meaning | Unit |
|---|---|---|
| x | Input value (must be positive) | dimensionless |
| b | Base of the logarithm | dimensionless |
| ln | Natural logarithm (base e) | dimensionless |
Worked Example
Problem
The pH of a solution is given by pH = −log₁₀[H⁺]. Find the pH if [H⁺] = 3.16 × 10⁻⁴ mol/L.
Solution
Step 1: pH = −log₁₀(3.16 × 10⁻⁴). Step 2: = −(log₁₀ 3.16 + log₁₀ 10⁻⁴) = −(0.5 − 4) = −(−3.5) = 3.5.
Answer
pH = 3.5 (acidic solution).
Properties of Logarithmic Functions f(x) = log_b(x)
| Property | b > 1 | 0 < b < 1 |
|---|---|---|
| Behaviour | Increasing | Decreasing |
| x-intercept | (1, 0) | (1, 0) |
| Vertical asymptote | x = 0 | x = 0 |
| Domain | (0, +∞) | (0, +∞) |
| Range | All real numbers | All real numbers |
Interactive Tools
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Related Terms
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.
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.
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.
Combines "logarithm" (from John Napier, 1614) with "function." The formal treatment of logarithmic functions as inverses of exponentials was developed by Euler in "Introductio in analysin infinitorum" (1748).