How do you solve ln x + ln(x-2) = 1?

1 Answer
Oct 17, 2015

1+sqrt(1+e)~~2.9283

Explanation:

Use the properties of logarithms to rewrite the expression.

ln(x(x-2))=1

Rewrite both sides in terms of the base e

e^(ln(x(x-2)))=e^1

From the properties of logarithms this becomes

x(x-2)=e

Distribute x on left hand side

x^2-2x=e

Subtract e form both sides

x^2-2x-e=0

Applying the quadratic formula

(-(-2)+-sqrt((-2)^2-4(1)(-e)))/2(1)

(2+-sqrt((4+4e)))/2

(2+-sqrt(4(1+e)))/2

2/2+-(2sqrt(1+e))/2

1+-sqrt(1+e)

We are only interested in 1+sqrt(1+e)~~2.9283

The other root is negative and not in the domain of the logarithmic function.