Question

What is the true solution to the logarithmic equation below? log2(6x)-log2(sqrtX)=2

A, x=0
B x=2/9
C. x=4/9
D. x=2/3

Answer 1

ANSWER

The true solution is


`x = (4)/(9)`
Step-by-step explanation

The logarithmic equation given to us is


`log_(2)(6x) - log_(2)( √(x) ) = 2`



We need to use the quotient rule of logarithms.


`log_(a)( M ) - log_(a)( N ) = log_(a)( ( M )/( N ) )`


When we apply this law the expression becomes


`log_(2)( (6x)/( √(x) ) ) = 2`


We now take the antilogarithm of both sides to get


`(6x)/( √(x) ) = {2}^(2)`




`(6x)/( √(x) ) = 4`


We square both sides to get,



`((6x)/( √(x) )) ^(2) = {4}^(2)`

We evaluate to obtain,


`\frac{36 {x}^(2) }{ x } = 16`



This simplifies to



`36x = 16`



We divide both sides by 36 to get


`x = (16)/(36)`
We simplify to get,

`x = (4)/(9)`