Solve for x

Problem

ln(x−8)−ln(x+7)=ln(x−10)−ln(x+8)

Solution

1. Identify the domain of the logarithmic expressions by ensuring all arguments are positive: x−8>0 x+7>0 x−10>0 and x+8>0 This implies x>10

2. Apply the quotient rule for logarithms, ln(a)−ln(b)=ln(a/b) to both sides of the equation.

ln((x−8)/(x+7))=ln((x−10)/(x+8))

3. Exponentiate both sides to remove the natural logs, using the property that if ln(A)=ln(B) then A=B

(x−8)/(x+7)=(x−10)/(x+8)

4. Cross-multiply to eliminate the fractions.

(x−8)*(x+8)=(x−10)*(x+7)

5. Expand both sides of the equation using the FOIL method.

x^2−64=x^2−3*x−70

6. Subtract x^2 from both sides to simplify the equation to a linear form.

−64=−3*x−70

7. Isolate the variable by adding 70 to both sides and then dividing by −3

6=−3*x

x=−2

8. Check the solution against the domain requirement x>10 Since x=−2 does not satisfy the domain, there is no real solution.

Final Answer

No real solution

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