Question

How do you solve `ln(x + 5) + ln(x - 1) = 2`?

Answer 1

`x=-2+sqrt(e^2+9)`

Explanation:

`ln(x+5)+ln(x−1)=2`
First notice the domain is:`x>1`, next:
Simplify the left side(LS) using: `log(a)+log(b)=log(ab)` :
`ln[(x+5)(x-1)]=2`
Expand and simplify inside the bracket on LS:
`ln(x^2+4x-5)=2`
Use the log property: `log_a(x)=yhArrx=a^y`:
`x^2+4x-5=e^2`
Solve the quadratic by completing the square:
`x^2+4x=e^2+5`
`x^2+4x+(4/2)^2=e^2+5+(4/2)^2`
Simplify:
`x^2+4x+4=e^2+5+4`
Or:
`(x+2)^2=e^2+9`
Take square root of both sides:
`x+2=+-sqrt(e^2+9)`
Subtract 2 from both sides:
`x=-2+-sqrt(e^2+9)`
Reject the negative root (not in the domain):
`x=-2+sqrt(e^2+9)`