Let's solve the equation [tex]\(4 + 5e^{x+2} = 11\)[/tex] step-by-step to determine which given choice is correct:
1. Isolate the exponential term:
Start by subtracting 4 from both sides of the equation:
[tex]\[
4 + 5e^{x+2} - 4 = 11 - 4
\][/tex]
Simplifying this, we have:
[tex]\[
5e^{x+2} = 7
\][/tex]
2. Solve for [tex]\(e^{x+2}\)[/tex]:
Next, we divide both sides by 5:
[tex]\[
e^{x+2} = \frac{7}{5}
\][/tex]
3. Take the natural logarithm of both sides:
Apply the natural logarithm ([tex]\(\ln\)[/tex]) to both sides to eliminate the exponential:
[tex]\[
\ln(e^{x+2}) = \ln\left(\frac{7}{5}\right)
\][/tex]
4. Simplify using the properties of logarithms:
Recall that [tex]\(\ln(e^y) = y\)[/tex]:
[tex]\[
x + 2 = \ln\left(\frac{7}{5}\right)
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Subtract 2 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \ln\left(\frac{7}{5}\right) - 2
\][/tex]
We have found that the solution to the equation [tex]\(4 + 5e^{x+2} = 11\)[/tex] is:
[tex]\[
x = \ln\left(\frac{7}{5}\right) - 2
\][/tex]
Therefore, among the provided choices, the correct one is:
[tex]\[
x = \ln\left(\frac{7}{5}\right) - 2
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\ln\left(\frac{7}{5}\right) - 2}
\][/tex]
