Log Calculator
Calculate logarithms in any base with our free online log calculator. Supports common log (base 10), natural log (base e), and custom bases. Get log values, antilog, and interactive log curve chart for positive real numbers.
About This Calculator
Our free online Log Calculator computes the logarithm of any positive real number in any base. Whether you need the common logarithm (log10), the natural logarithm (ln or logₑ), or a custom base like binary log (log2), this tool provides instant results with the change-of-base formula.
The logarithm is one of the most fundamental concepts in mathematics. It answers the question: to what power must a given base be raised to obtain a specific number? Formally, if b^y = x, then log_b(x) = y. This calculator applies the change-of-base formula: log_B(x) = ln(x) / ln(B), which works for any positive base B (except B = 1) and any positive x.
This calculator displays two key results: the log value (the exponent y) and the antilog (b^y, which equals the original number x). You can also view an interactive log curve chart that shows how the logarithmic function behaves across a range of positive x values.
Regional Notes
India (IN): Logarithms are taught extensively in CBSE and state board curricula from Class 11 onward. Common log tables with base 10 are still used in competitive exams. This calculator serves as a quick digital alternative for students and professionals.
United States (US): Logarithms are introduced in Algebra II and Precalculus courses. The natural logarithm (ln) is heavily used in AP Calculus, physics, and engineering. This calculator supports both common and natural logs for coursework and professional use.
United Kingdom (UK): Logarithms appear in A-Level Mathematics and Further Mathematics, particularly in the pure mathematics and statistics modules. The natural log is central to calculus topics. This tool helps students verify their work and explore log functions visually.
Frequently Asked Questions
What is a logarithm?
A logarithm answers the question: to what exponent must a given base be raised to produce a specific number? For example, log10(100) = 2 because 10^2 = 100. Logarithms are the inverse operation of exponentiation and are fundamental in mathematics, science, engineering, and finance.
What is the difference between common log and natural log?
The common logarithm (log10) uses base 10 and is widely used in science and engineering for representing orders of magnitude, pH scales, and decibel levels. The natural logarithm (ln or logₑ) uses base e (approximately 2.71828) and appears naturally in calculus, compound interest, population growth, and many physical laws.
How do you calculate logarithms in different bases?
Use the change-of-base formula: log_B(x) = ln(x) / ln(B). This calculator automatically applies this formula for any positive base except 1. Enter any positive number x, choose a base B (common bases: 10, e, 2), and the calculator returns log_B(x) instantly.
What is the antilog of a number?
The antilog is the inverse operation of a logarithm. If log_B(x) = y, then the antilog is B^y = x. For example, if log10(100) = 2, then antilog10(2) = 10^2 = 100. This calculator displays both the log value and its corresponding antilog so you can verify the result.
What are the domain restrictions for logarithmic functions?
Logarithms are only defined for positive real numbers (x > 0). The base must also be positive and cannot equal 1. This calculator validates inputs and will not compute for zero or negative numbers.
What are the practical applications of logarithms?
Logarithms appear everywhere: the Richter scale for earthquakes (base 10), pH in chemistry (base 10), decibels for sound intensity (base 10), compound interest and exponential growth (base e), binary logarithms (base 2) in computer science for algorithm analysis, population modeling, radioactive decay, and signal processing.