Question

Which is a solution for this equation? Log base 2 x = 2 - log base 2 (x - 3)

X = 1
X = 2
X = 3
X = 4
X = 5

Answer 1

Answer: Fourth Option

`x =4`

Explanation:

First we write the equation

`log_2(x) = 2- log_2(x-3)`

Now we use the properties of logarithms to simplify the expression

`log_2(x)+log_2(x-3) = 2`

The property of the sum of logarithms says that:

`log_a (B) + log_a (D) = log_a (B * D)`

Then

`log_2[x(x-3)]= 2`

Now use the property of the inverse of the logarithms

`a ^ {log_a (x)} = x`

`2^(log_2[(x)(x-3)])= 2^2`

`(x)(x-3))}= 4`

`x^2-3x -4=0`

`x^2-3x -4=(x-4)(x+1)=0`

Then the solution are

`x= -1` and
`x= 4`

We take the positive solution because the logarithm of a negative number does not exist

Finally the solution is:

`x =4`