Solve the equation.
-$$( \ln x ) ^ { 2 } - \ln x ^ { 5 } = - 6$$
A) $\left\{ e ^ { 2 } , e ^ { 3 } \right\} ; x \approx 7.3891 , x \approx 20.0855$
B) $\left\{ e ^ { - 2 } , e ^ { - 3 } \right\} ; x \approx 0.1353 , x \approx 0.0498$
C) $\{ 2,3 \}$
D) { }

Question

Solve the equation.
-$$( \ln x ) ^ { 2 } - \ln x ^ { 5 } = - 6$$
A) $\left\{ e ^ { 2 } , e ^ { 3 } \right\} ; x \approx 7.3891 , x \approx 20.0855$
B) $\left\{ e ^ { - 2 } , e ^ { - 3 } \right\} ; x \approx 0.1353 , x \approx 0.0498$
C) $\{ 2,3 \}$
D) {  }

Quizplus · Accepted Answer

First, simplify the equation using logarithm properties: $(\ln x)^2 - 5\ln x + 6 = 0$. This is a quadratic equation in terms of $\ln x$. Let $y = \ln x$, then the equation becomes $y^2 - 5y + 6 = 0$, which factors to $(y - 2)(y - 3) = 0$. Thus, $y = 2$ or $y = 3$, meaning $\ln x = 2$ or $\ln x = 3$. Solving for $x$ gives $x = e^2$ or $x = e^3$, which corresponds to the values in choice A.

Solve the Equation (ln⁡x)2−ln⁡x5=−6( \ln x ) ^ { 2 } - \ln x ^ { 5 } = - 6(lnx)2−lnx5=−6

Solve the Equation $( \ln x ) ^ { 2 } - \ln x ^ { 5 } = - 6$