Glossary
Argument (of a logarithm)
The number *c* in log_b(c) for which the logarithm is being calculated.
Example:
For log₂(16), the argument is 16, which is the number we want to express as a power of the base 2.
Base (of a logarithm)
The number *b* in log_b(c) that is raised to a power to obtain the argument *c*; it must be positive and not equal to 1.
Example:
In log₃(9), the base is 3, meaning we are looking for the power to which 3 must be raised to get 9.
Common Logarithm
A logarithm with base 10, typically written as log(c) without an explicit base.
Example:
If a sound has an intensity 1000 times a reference level, its decibel level uses a common logarithm, log(1000) = 3.
Euler's Number (e)
An irrational mathematical constant approximately equal to 2.71828, which is the base of the natural logarithm.
Example:
In continuous compound interest, the formula A = Pe^(rt) uses Euler's number to model exponential growth.
Inverse Functions
Functions that 'undo' each other; for logarithms and exponentials, if f(x) = b^x, then f⁻¹(x) = log_b(x).
Example:
Since f(x) = 2^x and g(x) = log₂(x) are inverse functions, if you calculate 2^3 = 8, then log₂(8) will bring you back to 3.
Linear Scale
A scale where units are equally spaced, representing a fixed increment between each mark.
Example:
A standard ruler uses a linear scale, where each centimeter or inch mark represents an equal distance.
Logarithm
The exponent to which a base must be raised to produce a given number; it is the answer to a logarithmic expression.
Example:
In log₅(25) = 2, the logarithm is 2, because 5 raised to the power of 2 equals 25.
Logarithmic Expression
An expression written as log_b(c) that asks 'To what power must I raise the base *b* to get *c*?'
Example:
When you see the logarithmic expression log₂(8), you're asking '2 to what power equals 8?' The answer is 3.
Logarithmic Scale
A scale where units represent a multiplicative change of the base, with each unit being a power of the base.
Example:
The Richter scale for earthquake magnitudes uses a logarithmic scale, where an increase of 1 point means a 10-fold increase in intensity.
Natural Logarithm
A logarithm with base *e* (Euler's number, approximately 2.71828), written as ln(c).
Example:
The growth of a bacterial population often follows an exponential model, and calculating the time it takes to reach a certain size might involve a natural logarithm, like ln(e⁵) = 5.