Logarithm Calculator
Calculate logarithms in any base — common (base 10), natural (base e), binary (base 2), or custom base.
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How to use this calculator
Any logarithm can be computed from the natural log using the change-of-base formula.
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Enter the number you want to take the logarithm of.
- 2
Select the base: base 10 (common log), base e (natural log), base 2 (binary), or custom.
- 3
For a custom base, enter it in the custom base field.
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The result shows your chosen log plus common/natural/binary logs for reference.
Frequently asked questions
What is a logarithm?
log_base(x) answers: "What power do I raise the base to, in order to get x?" For example, log₁₀(1000) = 3 because 10³ = 1000. It is the inverse of exponentiation.
What is the natural logarithm (ln)?
The natural logarithm uses base e ≈ 2.71828 (Euler's number). It appears naturally in compound interest, population growth, radioactive decay, and entropy. ln(e) = 1, ln(1) = 0, ln(e²) = 2.
Why is log base 2 important in computing?
Binary logarithm (log₂) tells how many bits are needed to represent a number. log₂(256) = 8, meaning 8 bits. In algorithms, O(log n) complexity typically means log₂ — the number of times you can halve n before reaching 1.
What are the key logarithm rules?
log(a×b) = log(a) + log(b); log(a/b) = log(a) − log(b); log(aⁿ) = n×log(a); log(1) = 0; log_a(a) = 1. These rules convert multiplication into addition, which is why logarithms were used for calculation before computers.
Logarithms — from slide rules to algorithms
How to use the logarithm
Use this logarithm to ogarithms in any base — common (base 10), natural (base e), binary (base 2), or custom base. Enter your values above and get your result in seconds. The tool is free, works on all devices, and keeps your data private — nothing is stored or shared.
How the logarithm works
The logarithm calculator uses standard formulas used in mathematics, science, and engineering. Enter your inputs, and the tool calculates the result instantly in your browser. No server-side processing means your data stays on your device. Results update in real time as you change inputs.
Why logarithms were invented
John Napier introduced logarithms in 1614 to simplify multiplication and division into addition and subtraction. Before calculators, astronomers and navigators used log tables to multiply large numbers. log(a×b) = log(a) + log(b) — so a table lookup and addition replaced manual multiplication.
Logarithmic scales in science
Many natural phenomena span many orders of magnitude and are measured on log scales. The Richter scale: a magnitude 7 earthquake is 10× more powerful than magnitude 6. Decibels (dB): each 10 dB increase is 10× the power. pH: each unit is 10× the hydrogen ion concentration.
Logarithms in computer science
Binary search halves the search space each step — O(log₂ n) comparisons. Balanced BSTs have O(log n) insert/search. Shannon's information entropy H = −Σ p log₂(p) measures bits of information. Huffman coding, compression, and cryptographic key sizes all rely on binary logarithms.
Logarithm: how it works
Built on well-established mathematical principles, this tool delivers accurate results for students, researchers, and professionals. Enter your values and get instant clarity without specialist software or manual arithmetic.
Who uses this tool?
Teachers, students, engineers, and data analysts use it to verify calculations, check homework, and solve problems faster. It is intuitive enough for beginners while comprehensive enough for professional use.
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