log_3x=log_9(6x) Solve the equation.

To solve the equation log_3(x)=log_9(6x), we may apply logarithm properties.
Apply the logarithm property: log_a(b)= (log_c(b))/log_c(a) on log_3(x) , we get:
(log_9(x))/(log_9(3))=log_9(6x)
Let 3 =sqrt(9) = 9^(1/2)
(log_9(x))/(log_9(9^(1/2)))=log_9(6x)
Apply the logarithm property: log(x^n)= n*log(x) and log_a(a)=1 on log_9(9^(1/2)) .
(log_9(x))/(1/2log_9(9))=log_9(6x)
(log_9(x))/(1/2*1)=log_9(6x)
(log_9(x))/(1/2)=log_9(6x)
log_9(x)*(2/1)=log_9(6x)
2log_9(x)=log_9(6x)
Apply the logarithm property: log(x*y)=log(x)+log(y) on log_9(6x) .
2log_9(x)=log_9(6)+log_9(x)
2log_9(x)-log_9(x)=log_9(6)
(2-1)log_9(x)=log_9(6)
log_9(x)=log_9(6)
Apply the logarithm property:a^(log_a(x))=x on both sides.
9^(log_9(x))=9^(log_9(6))
x=6
Check: Plug-in x=6 on log_3(x)=log_9(6x).
log_3(6)=?log_9(6*6)
log_3(6)=?log_9(36)
1.631~~1.631
Final Answer:
x=6 is a real solution for the equation log_3(x)=log_9(6x) .