Expert Answers
lemjay eNotes educator| Certified Educator


To solve, express left side as one logarithm.  To do so, apply the quotient property which is ln M - ln N = ln(M/N).

`ln((x-4)/(x+1))=ln 6`

Since both sides of the equation have the same base of logarithm, to solve for x set the arguments of ln equal to each other.

`(x-4)/(x+1) = 6`

Then, multiply both sides by x+1 to simplify the equation.



Then, bring together the terms with x on one side of the equation. To do so, subtract both sides by x.



Also, bring together the terms without x on one side of the equation. So, subtract both sides by 6.



And divide both sides by 5.



Hence, the solution is x=-2.

oldnick | Student

`ln(x -4) - ln(x +1) = ln6`

`ln[(x-4)/(x+1)]= ln6`

passing to the exponent



Now multiply both side by  (x +1)

`(x+1)(x-4)/(x+1)= 6(x+1)`


`x-4= 6x + 6`

adding both sides 4:

`x -4 + 4 = 6x + 6 +4`

`x = 6x+ 10`

subtracting now both sides 6x:

 `x - 6x = 6x - 6x + 10`


`x= -2`