Question 44

x = 3/4

To solve the equation 2ln x = ln(x+3) + ln(x-1), we can rewrite the right side using logarithmic properties, leading to the solution x = 3/4.

A) x = 2/3

This option is incorrect as substituting x = 2/3 into the original equation does not satisfy the equality. The left side becomes 2ln(2/3), while the right side results in ln(2/3 + 3) + ln(2/3 - 1), leading to an inconsistency.

B) x = 3/4

This option is correct. Substituting x = 3/4 into the original equation yields both sides equal, confirming that it satisfies the equation. The transformation of the equation supports this solution.

C) x = sqrt(3)

This option is incorrect. When substituting x = sqrt(3) into the equation, the values do not match on both sides. The left side does not equal the logarithm values resulting from the right side, indicating it is not a solution.

D) x = 1 + sqrt(3)

This option is also incorrect. Substituting x = 1 + sqrt(3) leads to a left side that diverges from the right side in the equality, which means it does not satisfy the original equation.

E) The equation has no solution.

This option is incorrect. The equation does indeed have a solution, which is x = 3/4, thus disproving the claim that there is no solution.

Conclusion

The solution x = 3/4 is definitive as it satisfies the equation when both sides are evaluated. The other options do not fulfill the conditions of the equation, either leading to contradictions or mismatched values, confirming that B is the only viable answer.