Logarithm
A logarithm is the exponent to which a specified base must be raised to produce a given number. It is the inverse operation of exponentiation.
Exponential function: A function in which the variable appears in the exponent, commonly written f(x) = aˣ. Logarithms reverse this operation, recovering the exponent from the function's output.
Exponentiation: The operation of raising a number to a power. Logarithms are defined by reversing this operation.
Decibel: A logarithmic unit that expresses ratios of power or amplitude, commonly used for sound and signal levels. Sound levels use logarithms to compress enormous power ratios into manageable numbers.
Common logarithm: A logarithm with base 10, written log(x) in many contexts. It is a familiar fixed-base case, distinct from the natural logarithm.
Product rule for logarithms: The identity logᵦ(xy) = logᵦ(x) + logᵦ(y), for positive x and y. It converts multiplication inside a logarithm into addition of logarithms.
Power: An expression formed by raising a base to an exponent, such as bˣ. A logarithm identifies the exponent in a power with a specified base.
Richter magnitude scale: A logarithmic scale originally used to assign magnitudes to earthquakes from seismograph readings. Its logarithmic structure represents large differences in seismic-wave amplitude compactly.
Binary logarithm: A logarithm with base 2, written log₂(x). It counts powers of two and is common in computing and information theory.
Change of base formula: The identity logᵦ(x) = logₐ(x) / logₐ(b), which expresses a logarithm using another base. It lets calculations use a convenient base without changing the value.
Real number: A number on the continuous number line, including rational and irrational numbers. Real logarithms usually take positive real inputs and produce real exponents.