Equation field guide
Logarithms: the exponent you need
A logarithm answers an inverse question about powers. log_b(x) is the exponent y for which bʸ = x.
The inverse of an exponent
If 10² = 100, then log₁₀(100) = 2. The base indicates the repeated multiplier. In scientific writing, log may mean base 10 or the natural logarithm; an article should define the convention.
The natural logarithm, written ln(x), has base e, approximately 2.718. It occurs often in continuous growth, decay, and calculus.
Domain and scale
For real-number logarithms, the input x must be positive, and the base must be positive and different from 1. There is no real log(0) or log of a negative number.
Logarithms turn multiplicative changes into additive distances. A tenfold increase adds 1 to a base-10 log. This is why logarithmic scales are useful when values span many orders of magnitude.
Reading log terms in formulas
log(a/b) = log(a) − log(b), and log(ab) = log(a) + log(b), provided the values are in the allowed domain. These rules often explain why a formula contains a sum of logs rather than a product.
Do not distribute a logarithm across addition: log(a + b) is not generally log(a) + log(b).