In order for
$Q^{\frac{1}{n_K}}$
to have a chance to look decent, TeX would have to provide not just two but, in fact, three subscript depth layers.
TeX's two basic subscript and superscript depth layers in math mode are called \scriptstyle, for a 1-0.7=30% linear reduction in font size relative to \normalsize, and \scriptscriptstyle, for a (1-0.7)^2 \approx 50% linear reduction in font size.
A third-level of subscripts and superscripts would have entailed performing a (1-0.7)^3 \approx 66% linear reduction in font size -- not infeasible from a purely technical point of view, I suppose, but definitely pushing (and quite likely exceeding) the boundaries of visual discernibility on paper and on screen for all but the highest-resolution devices.
Let's examine the formula $Q^{\frac{1}{n_K}}$ more closely: While Q is processed in \textstyle, 1 and n are processed in \scriptscriptstyle (because they occur in the numerator and denominator of a \frac expression). And, because TeX doesn't provide a third-level of subscript depth, K gets typeset in \scriptscriptstyle as well, making it look too large relative to n.
The fix? Abandon your poor approach to selecting notation for fraction-type material in an exponent. Specifically, replace the \frac notation with inline-fraction notation for the superscript material, i.e., write Q^{1/n_K} instead of Q^{\frac{1}{n_K}}.
The only meaningful, i.e., still somewhat decent-looking, alternative to writing Q^{1/n_K} would be to load the amsmath package (which provides the \tfrac macro) and write Q^{\tfrac{1}{n_K}}. However, this method risks creating a needlessly large, even turgid-looking, exponent term.
Note the optional use of \mkern-1.5mu (which corresponds to half of a negative thinspace, or \!) to "snug up" the K term to n.
\documentclass{article}
\usepackage{amsmath} % for '\tfrac' macro
\begin{document}
\[
Q^{\frac{1}{n_K}}
\quad
Q^{1/n_{\mkern-1.5mu K}}
\quad
Q^{\tfrac{1}{n_{\mkern-1.5mu K}}}
\]
\end{document}
Addendum to address the additional material posted by the OP, which asks how to best typeset an expression of the form
\frac{x^{1-\frac{c_1}{\log(Q(t+3)^{n_K})}}}{\log x}
I maintain that an inline-math expression would look better. Even better, though, would be to replace \frac{c_1}{\log(Q(t+3)^{n_K})} with a symbol such as, say, \kappa, and to state separately what \kappa is.
In the following screenshot, I try to make the version that employs the \frac expression as good as possible by employing \tfrac instead of \frac.
\documentclass{article}
\usepackage{amsmath} % for '\tfrac' macro
\usepackage{graphicx} % for '\scalebox' macro
\newcommand{\nK}{n_{\mkern-1.5mu K}} % snug up 'n' and 'K' terms
% create a de-facto 3rd-level-subscript form of 'K':
\newcommand{\smallK}{\scalebox{0.7}{$\scriptscriptstyle K$}}
\newcommand{\nKK}{n_{\mkern-1.5mu\smallK}}
\begin{document}
$\displaystyle
\frac{x^{1-\tfrac{c_1}{\log(Q(t+3)^{n_K})}}}{\log x} % note the use of '\tfrac`
\quad\text{vs.}\quad
x^{\{1-c_1/[\log(Q(t+3)^{\nKK})]\}}\big/\log x
$
\bigskip
$x^{1-\kappa}/\log x$, where $\kappa = c_1/[\log(Q(t+3)^{\nK})]$
\end{document}
n_KandQall have very particular meanings, and they can't be changed. Also the fractions are sometimes very long, and I can't really use/./" -- There's always a (notational) choice. If the fraction term in the exponent is very long, it's advisable to display it separately (especially as it's clearly important): Instead of$Q^{\frac{...}{...}$, I suggest you write$Q^{z}$, where $z=(...)/(...)$. That way, the important terms will be displayed in\textstylemath mode instead of\scriptscriptstylemath mode.