To solve the equation [tex]\( \log_{0.5}(x) = \log_3(2) + x \)[/tex], we can interpret it as two functions that should intersect at the solution of the given equation.
First, we analyze the left-hand side, [tex]\( \log_{0.5}(x) \)[/tex]. Using the change of base formula for logarithms, we can rewrite this as:
[tex]\[
\log_{0.5}(x) = \frac{\log(x)}{\log(0.5)}
\][/tex]
Thus, the function [tex]\( y_1 \)[/tex] is:
[tex]\[
y_1 = \frac{\log(x)}{\log(0.5)}
\][/tex]
Next, we look at the right-hand side of the given equation, [tex]\( \log_3(2) + x \)[/tex]. Again, using the change of base formula, [tex]\( \log_3(2) \)[/tex] can be rewritten as:
[tex]\[
\log_3(2) = \frac{\log(2)}{\log(3)}
\][/tex]
Thus, the equation simplifies to:
[tex]\[
y_2 = \frac{\log(2)}{\log(3)} + x
\][/tex]
So, the functions become:
[tex]\[
y_1 = \frac{\log(x)}{\log(0.5)}
\][/tex]
[tex]\[
y_2 = \frac{\log(2)}{\log(3)} + x
\][/tex]
Therefore, the correct system of equations that could be graphed to solve the given equation is:
[tex]\[
y_1 = \frac{\log(x)}{\log(0.5)}, \quad y_2 = \frac{\log(2)}{\log(3)} + x
\][/tex]
This matches perfectly with the choice #4 in the options provided:
[tex]\[
\text{Choice: } y_1 = \frac{\log x}{\log 0.5} \cdot y_2 = \frac{\log 2}{\log 3} + x
\][/tex]
Hence, the correct choice is:
[tex]\[
\boxed{4}
\][/tex]
