lnlny

lnlny denotes the natural logarithm of the natural logarithm of y, written as ln(ln(y)). It is defined for real-valued outputs when y > 1, since ln(y) must be positive to take its logarithm.

The function is strictly increasing on its domain. Its first derivative is f'(y) = 1/(y ln y). The second derivative is f''(y) = -(ln y + 1)/(y^2 (ln y)^2), which is negative for y > 1, so ln(ln y) is concave on this interval.

As y approaches 1 from above, ln y tends to 0+, and ln(ln y) tends to −∞. As y tends to infinity, ln y grows without bound, and ln(ln y) also grows without bound, but very slowly. Example values illustrate this behavior: lnln e = 0, lnln 10 ≈ 0.834, and lnln 2 ≈ −0.366. These illustrate that the function takes negative values for small y > 1 and crosses zero at y = e.

In mathematics and related fields, ln(ln y) is used to describe slow growth and appears in asymptotic analyses, integrals, and algorithms where double logarithms arise. In real-valued contexts, the domain is restricted to y > 1; extensions to complex numbers are possible but lie beyond the scope of the real-valued function. See also natural logarithm and the broader concept of iterated logarithms.