#1#. Start by moving all logs to the left side of the equation.
#log_3(x)=log_9(7x)-6#
#log_3(x)-log_9(7x)=-6#
#2#. Use the change of base formula, #log_color(blue)n(color(red)m)=(log_color(purple)b(color(red)m))/(log_color(purple)b(color(blue)n))#, to rewrite #log_9(7x)# with a base of #3#.
#log_3(x)-(log_3(7x))/(log_3(9))=-6#
#3#. Use the log property, #log_color(purple)b(color(purple)b^color(orange)x)=color(orange)x#, to rewrite #log_3(9)#.
#log_3(x)-(log_3(7x))/(log_3(3^2))=-6#
#log_3(x)-(log_3(7x))/2=-6#
#4#. Use the log property, #log_color(purple)b(color(red)m^color(blue)n)=color(blue)n*log_color(purple)b(color(red)m)#, to rewrite #(log_3(7x))/2#.
#log_3(x)-1/2(log_3(7x))=-6#
#log_3(x)-log_3((7x)^(1/2))=-6#
#5#. Use the log property, #log_color(purple)b(color(red)m/color(blue)n)=log_color(purple)b(color(red)m)-log_color(purple)b(color(blue)n)# to simplify the left side of the equation.
#log_3(x/((7x)^(1/2)))=-6#
#log_3(x^(1/2)/7^(1/2))=-6#
#6#. Use the log property, #log_color(purple)b(color(purple)b^color(orange)x)=color(orange)x#, to rewrite the right side of the equation.
#log_3(x^(1/2)/7^(1/2))=-log_3(3^6)#
#7#. Use the log property, #log_color(purple)b(color(red)m^color(blue)n)=color(blue)n*log_color(purple)b(color(red)m)#, to rewrite the right side of the equation.
#log_3(x^(1/2)/7^(1/2))=log_3(3^-6)#
#8#. Since the equation now follows a "#log=log#" situation, where the bases are the same on both sides, rewrite the equation without the "#log#" portion.
#x^(1/2)/7^(1/2)=3^-6#
#9#. Solve for #x#.
#x^(1/2)/7^(1/2)=1/3^6#
#x^(1/2)/7^(1/2)=1/729#
#x^(1/2)=7^(1/2)/729#
#(x^(1/2))^2=(7^(1/2)/729)^2#
#color(green)(|bar(ul(color(white)(a/a)x=7/531441color(white)(a/a)|)))#
