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`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)` .

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