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)|)))
