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

1 Answer
Sep 21, 2015

#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)#