[tex]If \ log_q(xy)=3, \ then \ log_qx+log_qy=3 \ . . . . . \ (1) \\ If \ log_q(x^2y^3)=4, \ then \ log_qx^2+log_qy^3=2log_qx+3log_qy=4 \ . . . . . \ (2) \\ From \ (1), \ log_qx=3-log_qy \\ substituting \ for \ log_qx \ in \ (2) \ gives: \\ 2(3-log_qy)+3log_qy=4 \\ 6-2log_qy+3log_qy=4 \\ \\ Therefore, \ log_qy=4-6=-2; \ and \\ log_qx=3-log_qy=3-(-2)=3+2=5[/tex]
