This paper investigates the wave equation with variable-exponent nonlinearity and logarithmic source term, given by the following: \[ u_{tt}-\Delta u+au_{t}|u_{t}|^{m(\cdot)-2}=bu|u|^{p(\cdot)-2}n|u|, \] where $a$ and $b$ are positive constants, and the functions $m(\cdot)$ and $p(\cdot)$ satisfy certain required conditions. Using the energy method and several inequality techniques, we establish a finite-time global nonexistence result for specific solutions with positive initial energy, under appropriate conditions. This type of equation has significant applications in various fields, including fluid dynamics, electrorheological fluids, quantum mechanics, nuclear physics, optics, and~geophysics.