what is x in ln (8-4x) -ln x^2 = ln 4

ln(8 - 4x) - ln(x²) = ln(4)

 

Using the property of logarithms that ln(a) - ln(b) = ln(a/b) we have

ln((8-4x)/x²) = ln(4)

 

This means we must have

(8 - 4x)/x² = 4

 

Multiply both sides by x²

8 - 4x = 4x²

 

Divide through by 4 and rearrange

x² + x - 2 = 0

 

This factorizes nicely as

(x - 1)*(x + 2) = 0

 

So the solutions are x = 1 and x = -2

ln(8-4x) - ln(x^2) = ln 4

 

(8-4x)ln - x^2ln= 4ln

 

-x^2ln + (8-4x)ln - 4ln = 0

 

ln(-x^2 + (8-4x) - 4) = 0

 

ln(-x^2 - 4x + 4) = 0

ln((-x - 2)(-x - 2)) = 0

 

ln(x) = 0 --> x = 1 since ln(1) = 0.

(-x - 2) = 0 --> -x = 2 --> x = -2.

 

x = 0.

x = -2.