To solve the equation [tex]\( 0 = \ln(x + 6) \)[/tex] for [tex]\( x \)[/tex] and round the answer to the nearest hundredth, follow these steps:
1. Start with the given equation:
[tex]\[
0 = \ln(x + 6)
\][/tex]
2. Isolate the logarithmic expression:
Since [tex]\(\ln(x + 6)\)[/tex] is already isolated, proceed to the next step.
3. Exponentiate both sides of the equation to remove the natural logarithm:
Recall that [tex]\( e^{\ln(y)} = y \)[/tex]. Exponentiating both sides gives:
[tex]\[
e^0 = e^{\ln(x + 6)}
\][/tex]
4. Simplify the equation:
[tex]\[
1 = x + 6
\][/tex]
This is because [tex]\( e^0 = 1 \)[/tex] and [tex]\( e^{\ln(y)} = y \)[/tex].
5. Solve for [tex]\( x \)[/tex]:
Subtract 6 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x = 1 - 6
\][/tex]
Simplifying this, we get:
[tex]\[
x = -5
\][/tex]
6. Round the answer to the nearest hundredth:
The value [tex]\(-5\)[/tex] is already an integer and does not have any decimal places, so it remains [tex]\(-5.00\)[/tex] when rounded to the nearest hundredth.
Therefore, the solution to the equation [tex]\( 0 = \ln(x + 6) \)[/tex] rounded to the nearest hundredth is:
[tex]\[
\boxed{-5.00}
\][/tex]
