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Question 41, 3.4.89
HW Score: $84.15\mathrm{\%},34.5$ of 41 points
Points: 0 of 1
Solve the logarithmic equation. Be sure to reject any value of $x$ that is not in the domain of the original logarithmic expressions. Give the exact answer.
$$\ln (x-2)+\ln (x+1)=\ln (x-3)$$

Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Step 1 ：First, combine the two logarithms on the left side of the equation using the property ln(a) + ln(b) = ln(ab). This gives ln((x-2)(x+1)) = ln(x-3).

Step 2 ：Next, equate the arguments of the logarithms, which gives the equation (x-2)(x+1) = x-3. Solve this equation for x.

Step 3 ：The solution to the equation (x-2)(x+1) = x-3 is x = 1.

Step 4 ：However, check if this solution is in the domain of the original logarithmic expressions. The domain of ln(x-2), ln(x+1), and ln(x-3) is x > 2, x > -1, and x > 3 respectively.

Step 5 ：Therefore, the solution x = 1 is not in the domain of the original logarithmic expressions and must be rejected.

Step 6 ：The final answer is $\boxed{{\displaystyle \mathrm{\varnothing }}}$, which means there are no solutions to the given logarithmic equation.