Natural logarithm (ln)
Calculate the natural logarithm (base e). With graph and explanation.
Co je natural logarithm
The natural logarithm ln x is the logarithm with base e (Euler’s number ≈ 2.71828). It appears wherever something changes continuously – growth, decay, interest. It is the most important logarithm in calculus.
Where it is used in practice
- Continuous compounding and growth – how long until an investment doubles.
- Radioactive decay and half-life in physics and chemistry.
- Statistics and probability (normal distribution, likelihood).
Good to know
The number e is the base of natural growth. We have ln e = 1 and ln 1 = 0. The derivative of ln x is 1/x, which makes it fundamental to integral calculus.
Logarithm and power are inverse operations
The logarithm is the inverse function of the power. Picturing it this way makes logarithms much easier to understand: logₐx = y ⟺ aʸ = x. For example log₂8 = 3, because 2³ = 8. The logarithm thus “unpacks” the exponent.
Rules of logarithms
In detail with examples: rules of logarithms.
Tip: a logarithm turns multiplication into addition – which is why logarithm tables and slide rules were once used to multiply large numbers quickly.