Log Calculator
Free online logarithm calculator supporting common log (base 10), natural log (base e), and custom bases. Get log values, antilog, and interactive log curve chart for any positive real number.
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 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: 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. The natural logarithm (ln or logₑ) uses base e (≈ 2.71828) and appears naturally in calculus, compound interest, and population growth. This calculator supports both plus any custom base.
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, and the calculator returns log_B(x) instantly along with the antilog.
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 since logarithm of zero is undefined and the logarithm of a negative number is not a real number.
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, and signal processing.
Why is the logarithm of 1 always 0?
The logarithm of 1 is always 0 regardless of the base because any positive number raised to the power of 0 equals 1. Mathematically, b⁰ = 1 for any b > 0, so log_b(1) = 0. This fundamental property holds for common log (log 1 = 0), natural log (ln 1 = 0), and any other base.
How do logarithms relate to exponential functions?
Logarithms and exponential functions are inverses of each other. If y = b^x is an exponential function, then x = log_b(y) is its logarithmic inverse. This means the graph of y = log_b(x) is the reflection of y = b^x across the line y = x. Understanding this relationship is key to solving exponential equations and modeling growth and decay.