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Math & Numbers Runs in browser Updated 30 Mar 2026

Logarithm Calculator

Calculate log base 10, natural log (ln), log base 2, or any custom base. Antilog mode and logarithm rules reference included.

Base: 10

Result

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Enter a value to see the result.

log(ab) = log(a) + log(b)Logarithm of a product
log(a/b) = log(a) − log(b)Logarithm of a quotient
log(aⁿ) = n × log(a)Logarithm of a power
logₐ(1) = 0Log of 1 is always 0
logₐ(a) = 1Log base a of a is always 1
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How to use Logarithm Calculator

  1. Choose a base

    Select log base 10 (common log), ln (base e), log base 2 (binary), or enter a custom base value.

  2. Enter a value

    Type the number you want to find the logarithm of. Must be positive.

  3. Read the result

    The logarithm appears instantly with the formula shown.

  4. Toggle antilog mode

    Switch to antilog mode to find x when you know the logarithm value — useful for reversing a log operation.

  5. Review the rules

    Expand the logarithm rules reference card to see product, quotient, power, and change-of-base rules.

Logarithm Calculator FAQ

What is a logarithm?
A logarithm answers: to what power must the base be raised to get a number? log_b(x) = y means b^y = x. Example: log₁₀(100) = 2 because 10² = 100.

How do I calculate a logarithm?
Enter the base and the number. The calculator uses the change-of-base formula internally: log_b(x) = ln(x) / ln(b).

What is the difference between log and ln?
log usually means base 10 (common logarithm). ln means base e (natural logarithm), where e ≈ 2.71828.

What is an antilog?
The antilog is the inverse operation. If log_b(x) = y, then antilog = b^y = x. For base 10: antilog(3) = 10³ = 1000.

What is the change-of-base formula?
log_b(x) = log(x) / log(b) = ln(x) / ln(b). This allows computing any base logarithm using common or natural log.

What are the logarithm product and quotient rules?
Product rule: log(a × b) = log(a) + log(b). Quotient rule: log(a/b) = log(a) − log(b). Power rule: log(a^n) = n × log(a).

What is log base 2 used for?
Log base 2 (binary logarithm) is used in information theory, computing, and algorithms. log₂(n) gives the number of bits needed to represent n values.

Can I calculate log of a negative number?
No. Logarithms are only defined for positive real numbers. The calculator displays an error for zero or negative inputs.

Can I use a custom base?
Yes. Enter any positive number other than 1 as the base and the calculator uses the change-of-base formula to compute the result.

Is any data sent to a server?
No. All calculations run entirely in your browser. Nothing is uploaded or stored.

Background

Compute logarithms in any base with full precision. Choose common log (base 10, written log), natural log (base e, written ln), binary log (base 2, written log₂), or enter any custom positive base. An antilog mode calculates the inverse operation — given y, find x where log_b(x)=y. A collapsible reference card lists the essential logarithm rules: product rule, quotient rule, power rule, change-of-base formula, and the relationship between log and ln. Useful for students in pre-calculus, calculus, chemistry (pH and reaction rates), physics (decibels, Richter scale), information theory (bits and entropy), and finance (compound growth). The change-of-base formula is used internally: log_b(x) = ln(x)/ln(b). All processing runs in your browser with no data uploaded.

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