log(x+2)-log(x)=3

Question

find the exact solution?

Answer 1

The equation is log (x + 2) - log (x ) = 3.

Apply quotient property logarithm : .

log [(x + 2)/x ] = 3

Apply exponential formula: log base e (x ) = a ⇔ x = e ^a.

(x + 2)/x = e ³

1 + (2/x ) =  e ³

2/x =  e ³ - 1

⇒ x = 2/(e ³ - 1)

Substitute the value e ³ = 20.08.

x = 2/(20.08 - 1)

   = 2/19.08

   = 0.104.

Exact solution of the equation is x = 0.104.