Answer :
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]
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]
