Must be a positive number (x > 0)
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Enter a value and click Calculate// compute log, ln, or any custom base in one click
Calculate natural log (ln), common log (log10), binary log (log2), or any custom base logarithm instantly. Free browser-based logarithm calculator.
Ready to calculate
Enter a value and click CalculateType any positive number in the input field โ decimals and scientific notation accepted.
Select ln (natural), logโโ (common), logโ (binary), or enter a custom base.
Click Calculate to see the result, step-by-step breakdown, and all standard logs at once.
A logarithm answers the question: "to what power must we raise the base to get this number?" This calculator handles all common logarithm types โ natural (e), common (base 10), binary (base 2), and any custom base you need.
A logarithm is the inverse of exponentiation. log_b(x) = y means b^y = x. For example, logโโ(100) = 2 because 10ยฒ = 100. Logarithms are used throughout math, science, and engineering.
The natural logarithm uses Euler's number e (โ 2.71828) as its base. ln(x) = log_e(x). It appears naturally in calculus, differential equations, and exponential growth models.
Log base 2 (binary logarithm) is fundamental in computer science. It tells you how many bits are needed to represent a number, and describes the complexity of binary search โ O(logโ n) operations.
No โ logarithms of negative numbers and zero are undefined in real number mathematics. You must enter a positive value (x > 0). Logarithms of negative numbers exist in complex number theory but are not covered here.
The antilog is the inverse operation. If log_b(x) = y, then the antilog is b^y = x. This calculator shows the antilog as a verification step โ it should equal your original input value.
Use the change-of-base formula: log_b(x) = ln(x) / ln(b) = log(x) / log(b). This calculator applies this formula automatically when you choose a custom base.
log_b(1) = 0 for any base b. This is because bโฐ = 1 for any positive b โ 1. It's a fundamental logarithm identity useful in many simplifications.
Results are computed using PHP's built-in math functions (IEEE 754 double precision), providing approximately 15-16 significant digits of accuracy. Results are displayed to 10 decimal places.
Our logarithm calculator is a fast, free browser-based tool that computes natural logarithms, common logarithms, binary logarithms, and logarithms of any custom base. Whether you're a student working through algebra, a developer analyzing algorithm complexity, or a scientist applying formulas, this tool provides instant, precise answers with step-by-step explanations.
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A logarithm answers a fundamental question: to what exponent must a given base be raised to produce a certain number? Written formally, log_b(x) = y means b^y = x. Logarithms are the mathematical inverse of exponentiation, much as subtraction is the inverse of addition.
Logarithms compress large ranges of numbers into manageable scales. The Richter scale for earthquakes, the decibel scale for sound, and the pH scale for acidity are all logarithmic. A magnitude 7 earthquake is not twice as powerful as a magnitude 6 โ it is ten times more powerful, because each step on the Richter scale represents a tenfold increase in amplitude.
Natural Logarithm (ln): Uses Euler's number e โ 2.71828 as the base. The natural log appears throughout calculus, statistics, and differential equations. It is the logarithm of choice in continuous growth models, including population growth, radioactive decay, and compound interest with continuous compounding. ln(e) = 1 by definition.
Common Logarithm (logโโ): Uses base 10 and is simply written as "log" in most textbooks. Before calculators existed, log tables were essential for multiplication and division of large numbers because log(a ร b) = log(a) + log(b). Common logs are still used extensively in chemistry (pH), acoustics (decibels), and signal processing.
Binary Logarithm (logโ): Uses base 2 and is the native language of computer science. The number of bits needed to represent n states is โlogโ(n)โ. Binary search halves the search space at each step, giving O(logโ n) time complexity. Entropy in information theory is measured in bits using logโ.
Custom Base Logarithms: Any positive number other than 1 can serve as a logarithm base. logโ(27) = 3 because 3ยณ = 27. Custom base logarithms use the change-of-base formula internally: log_b(x) = ln(x) / ln(b).
Understanding logarithm rules helps simplify complex expressions. The product rule states log(a ร b) = log(a) + log(b), turning multiplication into addition. The quotient rule states log(a / b) = log(a) โ log(b), turning division into subtraction. The power rule states log(a^n) = n ร log(a), pulling exponents out front. These three rules were the foundation of slide rules and logarithm tables, enabling complex calculations long before electronic computation.
Special values are worth memorizing: log_b(1) = 0 for any valid base (since bโฐ = 1), and log_b(b) = 1 for any valid base (since bยน = b). The logarithm of a number less than 1 but greater than 0 is negative for bases greater than 1.
Algorithm analysis relies heavily on logarithms. When an algorithm divides the problem size by 2 at each step โ like binary search, merge sort, or a balanced binary tree lookup โ the number of steps grows as logโ(n). For n = 1,000,000,000 (one billion), logโ(n) โ 30. This is why binary search through a billion records takes only about 30 comparisons.
Hash tables, heaps, B-trees, and segment trees all have logarithmic operations. Understanding logโ is essential for any serious study of algorithms and data structures.
The pH scale measures hydrogen ion concentration: pH = โlogโโ([Hโบ]). A pH of 7 is neutral, below 7 is acidic, above 7 is basic. Because it is logarithmic, a solution with pH 3 has 10,000 times more hydrogen ions than a solution with pH 7.
Sound intensity in decibels: dB = 10 ร logโโ(I / Iโ), where Iโ is the threshold of human hearing. Normal conversation at 60 dB represents a sound intensity 10^6 times greater than Iโ.
Stellar magnitude in astronomy uses a logarithmic scale where a difference of 5 magnitudes corresponds to a factor of 100 in brightness.
Finance: Continuous compound interest uses the natural logarithm. The time to double an investment at rate r is ln(2)/r, known as the natural doubling time.
Enter any positive real number in the input field โ decimals and very large or very small numbers work equally well. Select your desired base from the quick selector buttons: ln for natural log, logโโ for common log, logโ for binary log, or Custom to enter any base. Click Calculate to instantly see the result, a step-by-step breakdown showing how the answer was derived, and a summary of all three standard logarithms for your input value. The antilog verification confirms accuracy by computing the inverse operation.