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

Question

asked Oct 12, 2018 225k views

2 Answers

Grofit · Answer 1 · 2018-10-17T14:51:57+0000

Answer:

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