Solve $\ln x ^ { 2 } = 3$ for $x$.
A) $e ^ { 9 }$
B) $10 ^ { \sqrt { 3 } }$
C) $e ^ { \sqrt { 3 } }$
D) $- e ^ { 3 / 2 } , e ^ { 3 / 2 }$
E) no solution

Question

Quizplus · Accepted Answer

Using the property of logarithms, we have:
$$\ln x^2 = 2 \ln x = 3$$
$$\ln x = \frac{3}{2}$$
Taking the exponential of both sides, we get:
$$e^{\ln x} = e^{3/2}$$
$$x = e^{3/2}$$
However, we also need to check for the possibility of the negative solution since we squared the $x$. Plugging $-e^{3/2}$ into $\ln x ^ { 2 }$ gives us:
$$\ln (-e^{3/2}) ^2 = \ln e^3 = 3$$
Thus, $-e^{3/2}$ is also a valid solution. Therefore, the answer is D.

Solve ln⁡x2=3\ln x ^ { 2 } = 3lnx2=3 For xxx A)

Solve $\ln x ^ { 2 } = 3$ For $x$
A)