ln2x

ln2x denotes the natural logarithm of the argument 2x, i.e., ln(2x). It is defined for positive x, since 2x must be positive; thus x > 0. By the logarithm product rule, for x > 0 we have ln(2x) = ln 2 + ln x.

Key properties include its zero, derivative, and curvature. The function equals zero when 2x = 1, so at x = 1/2. The derivative is d/dx [ln(2x)] = 1/x for x > 0, by the chain rule. The second derivative is d2/dx2 [ln(2x)] = -1/x2, so the function is concave down on its domain.

Graphically, ln(2x) shares the shape of the natural logarithm ln x, but is shifted upward by ln 2, since ln(2x) = ln x + ln 2. There is a vertical asymptote at x = 0+; as x → 0+, ln(2x) → -∞, and as x → ∞, ln(2x) → ∞.

Applications and notes: ln(2x) is often used to simplify multiplicative relationships by converting them to additive ones, using ln(2x) = ln 2 + ln x. Its growth rate matches that of ln x, because the derivative is 1/x in both cases; the constant ln 2 only shifts the graph vertically and does not alter the slope.