To solve the equation [tex]\(4 + 5e^{x+2} = 11\)[/tex], follow these steps:
1. Isolate the exponential term:
[tex]\[
4 + 5e^{x+2} = 11 \implies 5e^{x+2} = 11 - 4 \implies 5e^{x+2} = 7
\][/tex]
2. Divide both sides by 5 to further isolate the exponential term:
[tex]\[
e^{x+2} = \frac{7}{5}
\][/tex]
3. Take the natural logarithm of both sides:
[tex]\[
\ln(e^{x+2}) = \ln\left(\frac{7}{5}\right)
\][/tex]
4. Use the property of logarithms [tex]\(\ln(e^y) = y\)[/tex] to simplify the left-hand side:
[tex]\[
x+2 = \ln\left(\frac{7}{5}\right)
\][/tex]
5. Isolate [tex]\(x\)[/tex] by subtracting 2 from both sides:
[tex]\[
x = \ln\left(\frac{7}{5}\right) - 2
\][/tex]
After following these steps, we find that the solution is:
[tex]\[
x = \ln\left(\frac{7}{5}\right) - 2
\][/tex]
Therefore, among the given choices, the correct answer is:
[tex]\[
x = \ln\left(\frac{7}{5}\right) - 2
\][/tex]
So, the correct option is:
[tex]\[
\boxed{x = \ln\left(\frac{7}{5}\right) - 2}
\][/tex]
