What is (lnx)^2?

The function (ln(x))^2, also written as (ln x)^2, represents the square of the natural logarithm of x. Here's some information about it:

• Definition: It is defined as (ln(x)) * (ln(x)), where ln(x) is the natural logarithm of x (logarithm to the base e).

• Domain: The domain of (ln(x))^2 is x > 0, because the natural logarithm function, ln(x), is only defined for positive values of x.

• Range: The range of (ln(x))^2 is [0, ∞). Since the square of any real number is non-negative, the output of the function will always be greater than or equal to zero. When x = 1, ln(x) = 0, and thus (ln(x))^2 = 0. As x approaches 0 from the right, ln(x) approaches -∞, and (ln(x))^2 approaches ∞. Similarly, as x approaches ∞, ln(x) approaches ∞, and (ln(x))^2 approaches ∞.

• Shape: The graph of (ln(x))^2 has a U-shape. It starts at ∞ as x approaches 0 from the right, decreases to 0 at x = 1, and then increases towards ∞ as x increases.

• Calculus:

  • Derivative: The derivative of (ln(x))^2 with respect to x is (2 ln(x))/x. This can be found using the chain rule.
  • Integral: The indefinite integral of (ln(x))^2 is x(ln(x))^2 - 2x ln(x) + 2x + C, where C is the constant of integration. Finding this integral requires integration by parts, twice.

• Properties: Because it involves a logarithmic function, it exhibits slow growth for large values of x, but its growth is eventually faster than ln(x).

• Applications: This function can appear in various mathematical contexts, especially in areas dealing with growth models, statistical distributions, and problems where logarithmic transformations are used.