You are analyzing exponential growth trends in quarterly sales data. To linearize the data model for your report, you must expand logarithmic expressions step-by-step. Recall the process for expanding and simplifying the expression $\log_4(2x^3y^2)$ by arranging the following steps in the correct order.

While auditing a software module that handles logarithmic data transformations, you encounter the expression $\log_4(2x^3y^2)$. To optimize the code, the expression must be replaced with its fully expanded and simplified form. Based on the standard properties of logarithms, which of the following is the correct expansion?

As a junior data analyst for a logistics firm, you are tasked with simplifying a cost-scaling formula represented by the expression $\log_4(2x^3y^2)$. To finalize your report for the operations manager, match each component of the original formula with its corresponding result in the final expanded and simplified form.

Documenting Logarithmic Transformations in Software Optimization

To linearize a growth trend for a quarterly sales report, an analyst expands the logarithmic expression $\log_4(2x^3y^2)$. In the final expanded and simplified form, the constant term $\log_4 2$ must be replaced by its numerical value. This simplified numerical value is ____.