11. Start by moving all logs to the left side of the equation.
log_3(x)=log_9(7x)-6log3(x)=log9(7x)−6
log_3(x)-log_9(7x)=-6log3(x)−log9(7x)=−6
22. 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))logn(m)=logb(m)logb(n), to rewrite log_9(7x)log9(7x) with a base of 33.
log_3(x)-(log_3(7x))/(log_3(9))=-6log3(x)−log3(7x)log3(9)=−6
33. Use the log property, log_color(purple)b(color(purple)b^color(orange)x)=color(orange)xlogb(bx)=x, to rewrite log_3(9)log3(9).
log_3(x)-(log_3(7x))/(log_3(3^2))=-6log3(x)−log3(7x)log3(32)=−6
log_3(x)-(log_3(7x))/2=-6log3(x)−log3(7x)2=−6
44. Use the log property, log_color(purple)b(color(red)m^color(blue)n)=color(blue)n*log_color(purple)b(color(red)m)logb(mn)=n⋅logb(m), to rewrite (log_3(7x))/2log3(7x)2.
log_3(x)-1/2(log_3(7x))=-6log3(x)−12(log3(7x))=−6
log_3(x)-log_3((7x)^(1/2))=-6log3(x)−log3((7x)12)=−6
55. 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)logb(mn)=logb(m)−logb(n) to simplify the left side of the equation.
log_3(x/((7x)^(1/2)))=-6log3(x(7x)12)=−6
log_3(x^(1/2)/7^(1/2))=-6log3(x12712)=−6
66. Use the log property, log_color(purple)b(color(purple)b^color(orange)x)=color(orange)xlogb(bx)=x, to rewrite the right side of the equation.
log_3(x^(1/2)/7^(1/2))=-log_3(3^6)log3(x12712)=−log3(36)
77. Use the log property, log_color(purple)b(color(red)m^color(blue)n)=color(blue)n*log_color(purple)b(color(red)m)logb(mn)=n⋅logb(m), to rewrite the right side of the equation.
log_3(x^(1/2)/7^(1/2))=log_3(3^-6)log3(x12712)=log3(3−6)
88. Since the equation now follows a "log=loglog=log" situation, where the bases are the same on both sides, rewrite the equation without the "loglog" portion.
x^(1/2)/7^(1/2)=3^-6x12712=3−6
99. Solve for xx.
x^(1/2)/7^(1/2)=1/3^6x12712=136
x^(1/2)/7^(1/2)=1/729x12712=1729
x^(1/2)=7^(1/2)/729x12=712729
(x^(1/2))^2=(7^(1/2)/729)^2(x12)2=(712729)2
color(green)(|bar(ul(color(white)(a/a)x=7/531441color(white)(a/a)|)))
