To solve the equation [tex]\(\ln(x) = -3\)[/tex], follow these steps:
1. Understand that the natural logarithm [tex]\(\ln(x)\)[/tex] is the logarithm to the base [tex]\(e\)[/tex] (where [tex]\(e \approx 2.718\)[/tex]).
2. To isolate [tex]\(x\)[/tex], exponentiate both sides of the equation using the base [tex]\(e\)[/tex], since [tex]\(e\)[/tex] and [tex]\(\ln\)[/tex] are inverse functions. This step transforms the equation into an exponential form:
[tex]\[
e^{\ln(x)} = e^{-3}
\][/tex]
3. Since [tex]\(e\)[/tex] and [tex]\(\ln\)[/tex] are inverse functions, [tex]\(e^{\ln(x)} = x\)[/tex]. Therefore, the equation simplifies to:
[tex]\[
x = e^{-3}
\][/tex]
4. Calculate [tex]\(e^{-3}\)[/tex]. The value of [tex]\(e^{-3}\)[/tex] is approximately [tex]\(0.049787068367863944\)[/tex].
5. Round the result to two decimal places:
[tex]\[
0.049787068367863944 \approx 0.05
\][/tex]
So, the solution to the equation [tex]\(\ln(x) = -3\)[/tex], rounded to two decimal places, is [tex]\(0.05\)[/tex].
Therefore, the correct answer is:
D. [tex]\(x = 0.05\)[/tex]
