Logarithm Calculator
Compute log₁₀(x), natural log ln(x), log of any base, or the antilog (inverse log) for any base. Customize mode shows all common bases at once and a step-by-step change-of-base derivation.
Where 1,000 sits on the log scale
Each tick is 10× the previous. Logarithms turn multiplication into addition — useful for sound (dB), earthquakes (Richter), pH, finance.
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How to use
- 1
Pick which logarithm you want: common (log₁₀), natural (ln), arbitrary base, or antilog.
- 2
Enter x. For arbitrary base or antilog, enter the base too.
- 3
Negative or zero x is undefined for log functions.
- 4
Switch to Customize to see log₂, log₁₀, and ln at once, plus step-by-step working using the change-of-base formula.
Frequently asked questions
log without a subscript usually means log base 10 (the common logarithm) in finance, engineering, and most calculators. ln is the natural logarithm with base e (≈ 2.71828), used in calculus, statistics, and continuously-compounded growth. They're related: log₁₀(x) = ln(x) / ln(10) ≈ ln(x) / 2.303.
logₐ(x) = ln(x) / ln(a) = log₁₀(x) / log₁₀(a). It lets you compute a log of any base using only natural logs or common logs — useful because most calculators only have ln and log₁₀ buttons. The Customize step-by-step shows this calculation.
There's no real number you can raise a positive base to and get a negative result. For example, 10^x is always positive — there's no x where 10^x = −5. Log of negative numbers is defined in complex analysis, but this calculator stays in the real numbers.
Antilog is the inverse of log. If log₁₀(x) = y, then antilog₁₀(y) = 10^y = x. It's useful when you have a logarithm and need the original value — common in chemistry (pH), acoustics (decibels), and seismology (Richter scale). Customize mode shows antilogs in base 2, 10, and e simultaneously.
No. Base 0 is undefined (0^x = 0 for any positive x, so log₀ has no meaningful inverse). Base 1 fails because 1 raised to any power is 1 — there's no unique exponent.
Product: logₐ(x·y) = logₐ(x) + logₐ(y). Quotient: logₐ(x/y) = logₐ(x) − logₐ(y). Power: logₐ(xⁿ) = n · logₐ(x). These turn multiplication into addition and exponents into multiplication — which is exactly why logarithms were invented (by Napier in 1614) before electronic calculators existed.
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Source: Standard logarithm definitions · Last verified 2026-06. This tool provides estimates only and is not legal, tax or financial advice. Always verify your specific situation with the relevant UAE authority or a licensed advisor before taking action.