Answer
Solution set = $\{23\}$.
Work Step by Step
$\log_{4}(3x-5)=3$
Logarithms are defined only for positive arguments, so any solution must satisfy
$3x-5 \gt 0\Rightarrow\qquad x \gt 5/3\qquad(*)$
Write the RHS as $\log_{4}($....$)$ using the basic property $\log_{b}b^{x}=x $
$\log_{4}(3x-5)=\log_{4}4^{3}$
logarithmic functions are one-to-one; if $\log_{b}M=\log_{b}N $, then M=N
$3x-5=4^{3}$
$ 3x-5=64\qquad $... add 5
$ 3x=69\qquad $... divide with 3
$ x=23$
which satisfies the condition (*), and is a valid solution.
Solution set = $\{23\}$.
