  Let $\psi:\mathbb{N}\to \mathbb{R}^+$ be a function satisfying $\phi(n)/n\to
\infty$ as $n \to \infty$. We investigate from a multifractal analysis point of
view the growth rate of the sums $\sum^n_{k=1}\log a_k(x)$ relative to
$\psi(n)$, where $[a_1(x),a_2(x), a_3(x)\cdots]$ denotes the continued fraction
expansion of $x\in (0,1)$. The upper (resp. lower) fast Khintchine spectrum is
defined by the Hausdorff dimension of the set of all points $x$ for which the
upper (resp. lower) limit of $\frac{1}{\psi(n)}\sum^n_{k=1}\log a_k(x)$ is $1$.
The precise formulas of these two spectra are completely determined, which
strengthens a result of Liao and Rams (2016).
