Sure, let’s solve the equation [tex]\(\ln \left(x^2-16\right)=0\)[/tex] step-by-step.
1. Understand the given equation:
[tex]\[\ln \left(x^2 - 16\right) = 0\][/tex]
2. Eliminate the natural logarithm by exponentiating both sides to remove the [tex]\(\ln\)[/tex]:
[tex]\[e^{\ln \left(x^2 - 16\right)} = e^0\][/tex]
Since [tex]\(e^{\ln y} = y\)[/tex] for any [tex]\(y > 0\)[/tex], this simplifies to:
[tex]\[x^2 - 16 = 1\][/tex]
3. Solve the resulting algebraic equation:
[tex]\[x^2 - 16 = 1\][/tex]
Add 16 to both sides:
[tex]\[x^2 = 17\][/tex]
4. Solve for [tex]\(x\)[/tex] by taking the square root of both sides:
[tex]\[x = \pm \sqrt{17}\][/tex]
Therefore, the solutions to the equation [tex]\(\ln \left(x^2 - 16\right) = 0\)[/tex] are:
[tex]\[x = \sqrt{17} \quad \text{and} \quad x = -\sqrt{17}\][/tex]
Thus, the correct option from the given choices is:
[tex]\[x = \pm \sqrt{17}\][/tex]
