Free Log Calculator
Calculate log base 10, natural log (ln), log base 2, and any custom base logarithm. Also computes the antilog (inverse). Free, private — all calculations run in your browser.
Same x = 100, Different Bases
Logarithm Rules Reference
About This Log Calculator
The Log Calculator computes logarithms in any base, including the three most commonly used: common logarithm (base 10), natural logarithm (base e), and binary logarithm (base 2). Enter any positive number and choose your base to get the logarithm and its inverse (antilog) instantly.
The Core Relationship
Logarithms are the inverse of exponentiation. Where exponentiation asks “what is aˢ?”, the logarithm asks “what power s gives me x when I raise a to it?” This inverse relationship is what makes logarithms invaluable for solving exponential equations and working with data that spans many orders of magnitude.
Assumptions and Limitations
- •Logarithms are only defined for positive real numbers — log(0) = −∞, log of negatives is undefined in real numbers
- •The base must be positive and not equal to 1 — log base 1 is undefined
- •For complex logarithms (log of negative numbers), a separate complex analysis calculator is needed
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When to Use This Calculator
Evaluate logarithmic expressions, verify base-change calculations, and compute complex log values for algebra, precalculus, and calculus problems.
Calculate pH = −log[H⁺] or [H⁺] = 10^(−pH). One of the most common logarithm applications in science education and lab work.
Calculate frequency ratios between musical notes (all based on log₂) and sound intensity levels in decibels (log₁₀-based).
Calculate algorithm complexity (O(log n) for binary search) and compute log₂ values for information theory, compression, and data structures.
Use ln to solve for time in compound interest problems: t = ln(A/P) / (r × n). Logarithms are the key to reversing exponential growth equations.
💡 Pro Tips
The change of base formula — logₐ(x) = log(x)/log(a) = ln(x)/ln(a) — is the most practical formula in logarithms. Most calculators only have log₁₀ and ln buttons, so any other base is computed via change of base. Memorise this formula and you can compute any logarithm on any scientific calculator.
The three logarithm rules (product, quotient, power) are the foundation of logarithmic simplification. The product rule is why dB (decibels) add arithmetically even though power ratios multiply: 40 dB + 30 dB = 70 dB corresponds to power ratio 10,000 × 1,000 = 10,000,000 (since log of products = sum of logs).
In chemistry, pH = −log₁₀[H⁺]. A pH change of 1 represents a 10× change in hydrogen ion concentration. A pH of 3 is 10× more acidic than pH 4, and 100× more acidic than pH 5. This is why logarithmic scales like pH, dB, and Richter are necessary — they make enormous differences in magnitude perceptible on a human scale.
Natural logarithm (ln) and exponential growth are inverses: ln(eˣ) = x and e^ln(x) = x. This makes ln the natural tool for solving any problem involving exponential growth or decay: half-life, population growth, radioactive decay, compound interest, and Newton's law of cooling. If you see a problem with e or eˣ, reach for ln.
Frequently Asked Questions
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