Answer:
x = 6
Step-by-step explanation:
Given : \(2\:log_2\:2\:+\:2\:log_26−\:log_2\:3x\:=\:3\)
We have to solve the given expression \(2\:log_2\:2\:+\:2\:log_26−\:log_2\:3x\:=\:3\)
Subtract \(2\log _2\left(2\right)+2\log _2\left(6\right)\) both sides , we have,
\(2\:log_2\:2\:+\:2\:log_26-\:log_2\:3x-(2\log _2\left(2\right)+2\log _2\left(6\right)):=\:3-(2\log _2\left(2\right)+2\log _2\left(6\right))\)
Simplify, we have,
\(\log _2\left(3x\right)=3-2\log _2\left(2\right)-2\log _2\left(6\right)\)
Divide both side by -1, we have,
\(\frac{-\log _2\left(3x\right)}{-1}=\frac{3}{-1}-\frac{2\log _2\left(2\right)}{-1}-\frac{2\log _2\left(6\right)}{-1}\)
Simplify, we have,
\(\log _2\left(3x\right)=-3+2\log _2\left(2\right)+2\log _2\left(6\right)\)
Apply log rule, \(a=\log _b\left(b^a\right)\)
\(2\log _2\left(6\right)-1=\log _2\left(2^{2\log _2\left(6\right)-1}\right)=\log _2\left(18\right)\)
When log have same base,
\(\log _b\left(f\left(x\right)\right)=\log _b\left(g\left(x\right)\right)\quad \Rightarrow \quad f\left(x\right)=g\left(x\right)\)
\(\mathrm{For\:}\log _2\left(3x\right)=\log _2\left(18\right)\mathrm{,\:\quad solve\:}3x=18\)
3x = 18
x = 6
