Solve for x

Problem

ln(x−4)−ln(x+1)=ln(6)

Solution

1. Apply the quotient rule for logarithms to the left side of the equation, which states ln(a)−ln(b)=ln(a/b)

ln((x−4)/(x+1))=ln(6)

2. Exponentiate both sides using the base e to remove the natural logarithms, since the function ƒ(x)=ln(x) is one-to-one.

(x−4)/(x+1)=6

3. Clear the fraction by multiplying both sides of the equation by the denominator (x+1)

x−4=6*(x+1)

4. Distribute the constant on the right side.

x−4=6*x+6

5. Isolate the variable x by subtracting x from both sides and subtracting 6 from both sides.

−10=5*x

6. Solve for x by dividing both sides by 5

x=−2

7. Check for extraneous solutions by substituting x=−2 back into the original logarithmic expressions. Since ln(−2−4)=ln(−6) and the domain of ln(x) is x>0 the value x=−2 is undefined.

No real solution

Final Answer

ln(x−4)−ln(x+1)=ln(6)⇒No real solution

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