Logarithm Calculator
The logarithm of a number in any base, plus the natural log.
Result
log_b(x)
3.0000
Natural log (ln)
6.9078
How it works
A logarithm answers the reverse of an exponent question: log_b(x) asks "to what power must I raise b to get x?". Since 10³ = 1000, log₁₀(1000) = 3. This calculator returns the log in your chosen base plus the natural log ln (base e ≈ 2.718), using the change-of-base rule log_b(x) = ln(x) ÷ ln(b). Logarithms compress huge ranges into small numbers, which is why they run the scales we use for extremes: pH in chemistry, decibels in sound, the Richter scale for earthquakes — one step on the scale is a multiplication underneath. Only positive numbers have (real) logarithms: b^anything is always positive, so log(0) and log of a negative number are undefined. Values between 0 and 1 have negative logs, and log_b(1) = 0 in every base.
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Frequently asked questions
What is a natural log?
The logarithm in base e (≈ 2.718), written ln, common in maths and science.
What base should I use?
Base 10 for everyday use, base e for calculus, base 2 in computing — set whichever you need.
Why can't I take the logarithm of zero or a negative number?
Because no real exponent turns a positive base into zero or a negative result — b^x is always positive. That leaves log(0) and log(−x) undefined in real numbers.
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