The mathematical property associated with ln x + ln y is the product rule of natural logarithms, expressed as ln(x ? y) = ln(x) + ln(y). The rule is used for adding together any two logarithm expressions that are to the same base, which for the natural logarithm is the base e.
For example, the expression ln(2) + ln(3) uses the product rule to become the ln(2 * 3), which is equal to the ln(6). The quotient rule for natural logarithms is related to the product rule and is used when a natural logarithm is subtracted from another natural logarithm. This rule is expressed as ln(x / y) = ln(x) - ln(y).
Other rules used to evaluate natural logarithm expressions that include both an x and y variable include the power rule, which is expressed in mathematical notation as ln(x^y) = y ? ln(x). Additional general identities used when evaluating natural logarithms include the derivative, which for the function f(x) = ln(x) is always equal to f'(x) = 1/x, and the integral, which for the same function is x ? (ln(x) - 1) + C. When the value for x in ln(x) is less than or equal to zero, the value is undefined. The natural logarithm of one is equal to zero. The limit of ln(x) as x approaches infinity is defined as infinity.