1/3log_5(12x)=2 Solve the equation. Check for extraneous solutions.

To evaluate the given equation 1/3log_5(12x)=2 , we may apply logarithm property: n* log_b(x) = log_b(x^n) .
log_5((12x)^(1/3))=2
Take the "log" on both sides to be able to apply the logarithm property: a^(log_a(x))=x .
5^(log_5((12x)^(1/3)))=5^(2)
(12x)^(1/3)= 25
Cubed both sides to cancel out the fractional exponent.
((12x)^(1/3))^3= (25)^3
(12x)^(1/3*3)=15625
(12x)^(3/3)=15625
12x =15625
Divide both sides by 12 .
(12x)/12 =(15625)/12
x =(15625)/12
Checking: Plug-in x=(15625)/12 on 1/3log_5(12x)=2 
1/3log_5(12*(15625)/12)=?2
1/3log_5(15625)=?2
log_5(15625^(1/3))=?2
log_5(root(3)(15625))=?2
log_5(25)=?2
log_5(5^2)=?2
2log_5(5)=?2
2*1=?2
2=2  TRUE

There is no extraneous solution. The x=(15625)/12 is a real solution for the given equation 1/3log_5(12x)=2 .

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