2log2x

2log2x denotes two times the logarithm of x with base 2, typically written as 2 log_2 x or 2 log_2(x). The expression is defined for x > 0. The base 2 makes log_2 x the inverse of the exponential function 2^x on positive x, and the function is increasing on its domain.

Properties and interpretation: By the power rule of logarithms, 2 log_2 x = log_2 (x^2). This identity is often used to simplify expressions or solve equations. The change-of-base formula gives log_2 x = ln x / ln 2, so 2 log_2 x = (2 ln x) / ln 2.

Examples: If x = 4, log_2 4 = 2, so 2 log_2 4 = 4. If x = 1/2, log_2 (1/2) = -1, so 2 log_2 (1/2) = -2. If x = 1, the value is 0.

Domain, range, and behavior: The domain is x > 0. As x approaches 0 from the right, log_2 x tends to -∞, and as x grows, log_2 x tends to ∞; multiplying by 2 does not change the range, which remains all real numbers.

Ambiguity note: Some readers might misread the expression as log_2(2x), which equals 1 + log_2 x and is distinct from 2 log_2 x. When clarity is needed, parentheses help avoid misinterpretation.

See also: logarithm, base change, exponential function.