Logarithm
Log properties, equations
Logarithms are the inverse operation of exponentiation. While exponents ask 'what is the result of repeated multiplication?', logarithms ask 'how many times must we multiply to get this result?' The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Logarithms simplify complex calculations involving multiplication, division, and powers, and are essential in many areas of mathematics, science, and engineering.
Key formulas and rules
Key concepts
Understanding Logarithm Notation
In log_b(a), 'b' is called the base and 'a' is called the argument or antilogarithm. Common bases include 10 (common logarithm, written as log or lg) and e ~= 2.718 (natural logarithm, written as ln). The base must be positive and not equal to 1, and the argument must be positive. Logarithms are only defined for positive real numbers.
Fundamental Logarithm Laws
The three fundamental laws are: (1) Product Rule - log of a product equals sum of logs, (2) Quotient Rule - log of a quotient equals difference of logs, and (3) Power Rule - log of a power equals exponent times log of base. These laws allow us to break down complex expressions into simpler components and are the foundation of all logarithmic calculations.