A delayed nonlinear wave equation with variable exponents of logarithmic type is discussed in this paper. In the presence of the logarithmic nonlinear source, we established a global existence result under sufficient conditions on the initial data only without imposing the Sobolev Logarithmic Inequality. After that, we established global results of exponential and polynomial types according to the range values of the exponents. At the end, we give a numerical study that supports our theoretical results.