lnxln10

lnxln10 denotes the product of the natural logarithm of x and the natural logarithm of 10: f(x) = ln(x) · ln(10). It is a real-valued function defined for x > 0.

Domain and basic behavior: Since ln(10) is a positive constant, f is defined for x > 0 and is strictly increasing on (0, ∞) because f′(x) = (ln 10)/x > 0. As x → 0+, ln x → −∞, so f(x) → −∞; as x → ∞, f(x) → ∞.

Relationship to base-10 logarithm: Using ln x = log10(x) · ln 10, we have f(x) = (ln 10)^2 · log10(x). Thus lnxln10 can also be viewed as a constant multiple of the base-10 logarithm.

Special values: f(1) = 0; f(e) = ln(10) ≈ 2.302585; f(10) = (ln 10)^2 ≈ 5.301898.

Differentiation and integration: f′(x) = (ln 10)/x; f″(x) = −(ln 10)/x^2. An antiderivative is ∫ f(x) dx = (ln 10)(x ln x − x) + C.

Note: The expression lnxln10 is not a standard named function but a simple product of two logarithms. Its utility arises in algebraic manipulations and calculus when logarithmic constants are factored or when converting between logarithm bases.