I found the following definition of equivalence to leading exponential order in the book
Mezard, Marc, and Andrea Montanari. Information, physics, and computation. Oxford University Press, 2009.
The definition is as follows:
The notation $A_N \dot{=} B_N$ will be used throughout the book to denote that two quantities $A_N$ and $B_N$ (which behave exponentially in $N$) are equal to leading exponential order, meaning $\lim_{N\rightarrow\infty} (1/N)\log(A_N/B_N) = 0$.
This is the first time I encounter this definition. Is it in common use somewhere else? Where can I study more about its properties?