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

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.

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

`x-4=6x+6`

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

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

`-4=5x+6`

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

`-4-6=5x+6-6`

`-10=5x`

And divide both sides by 5.

`(-10)/5=(5x)/5`

`-2=x` **Hence, the solution is x=-2.**

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

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

passing to the exponent

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

Now multiply both side by (x +1)

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

devleoping:

`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`

`-5x=10`

`x= -2`

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