Logarithm calculator
A logarithm answers the question of what power a base must be raised to in order to give a number: log_b(x) = y means that b to the power y equals x. The common logarithm has base 10, the natural logarithm has base e, roughly 2.71828, and the binary logarithm has base 2.
- Natural log ln(x)
- 6.90775528
- Common log log₁₀(x)
- 3
- Binary log log₂(x)
- 9.96578428
- Check: b to that power
- 1,000
Rates and thresholds
- Definition
- log_b(x) = y ⟺ bʸ = x
- Change of base
- log_b(x) = ln(x) / ln(b)
- Logarithm of a product
- log(xy) = log x + log y
- Logarithm of a power
- log(xⁿ) = n · log x
- The number e
- ≈ 2.718281828
What is the difference between log and ln?
ln always means the natural logarithm, base e, roughly 2.71828. What log means depends on context: in mathematics and engineering the base is usually 10, in computer science it is often 2, and in many programming languages the log function is in fact the natural logarithm. The safe habit is to write the base out — log₁₀ or log₂.
Why can't you take the logarithm of a negative number?
A positive base raised to any real power is always positive — 10 to any power never gives zero or a negative number. So the logarithm of zero or of a negative number is undefined over the reals. As the input approaches zero the logarithm falls without limit: log₁₀(0.001) = −3, log₁₀(0.000001) = −6.
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