Monotonicity and logarithmic concavity of two functions involving exponential function

The function for x > 0 is proved to be strictly decreasing. As an application of this monotonicity, the logarithmic concavity of the function for a ∈ ℝ and t ∈ (0, ∞) is verified. The possible origin and background of the function (*) are revealed to be related to the remainder of Binet's formula. Some applications of above results to the difference of θ(x) are noted.