Logarithm Calculator Calculate logarithms in any base (natural log, log₁₀, log₂, custom) and antilogarithms.

What Is Logarithm Calculator?

The Logarithm Calculator computes logarithms in any base and antilogarithms (inverse logarithms). Choose from natural log (base e), common log (base 10), binary log (base 2), or enter any custom base. In logarithm mode, find log_b(x) — the power to which base b must be raised to equal x. In antilogarithm mode, find b^x — the result of raising the base to the given exponent. The calculator simultaneously shows results in all common bases (ln, log₁₀, log₂) for easy reference.

Why Use Logarithm Calculator?

• Supports any base: e (ln), 10, 2, or custom
• Both logarithm and antilogarithm modes
• Shows results in all common bases simultaneously
• Displays the change-of-base formula

Common Use Cases

Mathematics

Solve logarithmic and exponential equations.

Science

Calculate pH (log₁₀), decibels, and Richter scale values.

Computer Science

Binary logarithms for algorithm complexity analysis.

Finance

Logarithmic returns and compound growth calculations.

Technical Guide

Logarithm definition: log_b(x) = y means b^y = x. The change of base formula: log_b(x) = ln(x) / ln(b). Key identities: log_b(1) = 0, log_b(b) = 1, log_b(x×y) = log_b(x) + log_b(y), log_b(x/y) = log_b(x) − log_b(y), log_b(x^n) = n × log_b(x). The natural logarithm (ln) uses base e ≈ 2.71828. Common applications: pH = −log₁₀([H⁺]), decibels = 10 × log₁₀(P₁/P₂), information bits = log₂(N), Richter magnitude uses log₁₀ of seismograph amplitude.

Tips & Best Practices

• 1
  log_b(x) is only defined for x > 0 and b > 0, b ≠ 1
• 2
  ln(e) = 1, log₁₀(10) = 1, log₂(2) = 1 — the log of the base always equals 1
• 3
  Negative logs mean the value is between 0 and 1 (e.g., log₁₀(0.1) = -1)
• 4
  The change of base formula lets you compute any log using ln: log_b(x) = ln(x)/ln(b)

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Frequently Asked Questions

Q What is a logarithm?

A logarithm answers: "to what power must the base be raised to get this number?" log₁₀(1000) = 3 because 10³ = 1000.

Q What is the natural logarithm?

The natural logarithm (ln) uses base e ≈ 2.71828. It appears naturally in calculus, continuous growth, and many physical phenomena.

Q Can you take the log of a negative number?

In real numbers, no. Logarithms are only defined for positive numbers. In complex numbers, logs of negative numbers involve imaginary components.

Q What is an antilogarithm?

The antilogarithm is the inverse operation: antilog_b(x) = b^x. If log₁₀(1000) = 3, then antilog₁₀(3) = 10³ = 1000.

Q Why is log₂ important in computer science?

Binary logarithms represent the number of bits needed to represent N values (log₂(N)) and appear in algorithm complexity analysis (e.g., binary search is O(log₂ n)).

About This Tool

Logarithm Calculator is a free online tool by FreeToolkit.ai. All processing happens directly in your browser — your data never leaves your device. No registration or installation required.