<s>[INST] $\log_{3}\frac{16^{x-3}+14}{4^{x-3}+2}=243$ , find the value of $x$. $\log_{3}\frac{16^{x-3}+14}{4^{x-3}+2}=243=3^5$
$\Rightarrow \frac{16^{x-3}+14}{4^{x-3}+2}= {3^3}^5=3^{243}$
I am not able to proceed from here. Please help !!!
The options given for this problem are :-

*

*$x$ is a rational number.

*$x$ is a natural number.

*$x$ is an even natural number.

*$x$ is a rational number less than $0$.

Thanks in advance !!!
 [/INST] Note that, for $x = 5$, the LHS of $\frac{16^{x-3}+14}{4^{x-3}+2}$ becomes $\frac{256+14}{16+2}=15$, with the LHS being less than $15$ for $x \lt 5$ (with $\lim_{x\to -\infty}\frac{16^{x-3}+14}{4^{x-3}+2}=\frac{14}{2} = 7$). Thus, $x \gt 5$. Also, we have that
$$\begin{equation}\begin{aligned}
\frac{16^{x-3}+14}{4^{x-3}+2} & = \frac{16(16^{x-4})+14}{4(4^{x-4})+2} \\
& = \frac{8(16^{x-4})+7}{2(4^{x-4})+1}
\end{aligned}\end{equation}\tag{1}\label{eq1A}$$
If $x$ is an integer, then since $8(16^{x-4})+7 \equiv 7 \pmod{8}$ and $2(4^{x-4})+1 \equiv 1 \pmod{8}$, the fraction in \eqref{eq1A} also being an integer (i.e., $3^{243}$) means it's congruent to $7$ modulo $8$. However, since $3^2 \equiv 1 \pmod{8}$, we instead have $3^{243} \equiv 3(3^{242}) \equiv 3(3^2)^{121} \equiv 3 \pmod{8}$.
This contradiction shows $x$ can't be an integer. Thus, it must either be a positive irrational or a non-integer rational. Since none of the provided second to fourth options are valid possibilities, this means the first option must be the correct one, i.e., $x$ is a non-integral rational number.
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