To solve the equation [tex]\(2 \ln (x-3) = -4\)[/tex], we need to follow these steps:
1. Isolate the natural logarithm (ln) term:
[tex]\[
2 \ln (x-3) = -4
\][/tex]
Divide both sides by 2:
[tex]\[
\ln (x-3) = -2
\][/tex]
2. Exponentiate both sides to eliminate the natural logarithm:
Recall that [tex]\(e^{\ln a} = a\)[/tex]. Thus, applying the exponential function to both sides:
[tex]\[
e^{\ln (x-3)} = e^{-2}
\][/tex]
[tex]\[
x-3 = e^{-2}
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 3 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x = e^{-2} + 3
\][/tex]
Now to compute the numerical value:
[tex]\[
e^{-2} \approx 0.1353352832366127
\][/tex]
Then,
[tex]\[
x = 0.1353352832366127 + 3
\][/tex]
[tex]\[
x \approx 3.135335283236613
\][/tex]
4. Round the final answer to the nearest hundredth:
[tex]\[
x \approx 3.14
\][/tex]
Therefore, the solution to the equation [tex]\(2 \ln (x-3) = -4\)[/tex] rounded to the nearest hundredth is:
[tex]\[
x \approx 3.14
\][/tex]
