Question

Solve 2 log2 2 + 2 log2 6 − log2 3x = 3.

Answer 1

log(base2)[2² * 6² / 3x] = 3
144 / 3x = 2^3 = 8
144/8 = 3x
18 = 3x
x = 6

Answer 2

x = 6

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,

`(-\log _2\left(3x\right))/(-1)=(3)/(-1)-(2\log _2\left(2\right))/(-1)-(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