ln1x1x

ln1x1x is not a standard mathematical symbol. In informal plain-text notation it is sometimes used to denote the natural logarithm of the product 1 × x × 1 × x. When read in this way, the expression simplifies to ln(x^2). In formal mathematics one would write ln(x^2) or, equivalently, 2 ln|x| for real x.

Domain and basic properties: If the interpretation is ln(x^2), the expression is defined for all real x except x = 0, since x^2 > 0 for x ≠ 0. The identity ln(x^2) = 2 ln|x| holds for x ≠ 0. The function is even, f(-x) = f(x), and it tends to -∞ as x approaches 0 from either side and grows without bound as |x| increases.

Examples: For x = 2, ln1x1x ≈ ln(4) ≈ 1.386. For x = -3, ln1x1x ≈ ln(9) ≈ 2.197. These illustrate the equivalence to ln(x^2).

Ambiguity and caution: Because ln1x1x is not standard notation, different calculators or programming languages may parse it differently, sometimes yielding 0 or producing a syntax error. To avoid ambiguity, use explicit notation like ln(x^2) or 2 ln|x| and include parentheses.

See also: natural logarithm, logarithm properties, logarithm of powers, absolute value inside logarithm.