Let's solve the equation [tex]\( 3 + 4 e^{x+1} = 11 \)[/tex] step-by-step and identify which of the given choices matches the solution.
1. Start with the given equation:
[tex]\[
3 + 4 e^{x+1} = 11
\][/tex]
2. Isolate the exponential term:
[tex]\[
4 e^{x+1} = 11 - 3
\][/tex]
[tex]\[
4 e^{x+1} = 8
\][/tex]
3. Divide both sides by 4 to further isolate the exponential term:
[tex]\[
e^{x+1} = \frac{8}{4}
\][/tex]
[tex]\[
e^{x+1} = 2
\][/tex]
4. Take the natural logarithm on both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
\ln(e^{x+1}) = \ln(2)
\][/tex]
5. Simplify using the property of logarithms [tex]\( \ln(e^y) = y \)[/tex]:
[tex]\[
x + 1 = \ln(2)
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \ln(2) - 1
\][/tex]
So, the solution to the equation [tex]\( 3 + 4 e^{x+1} = 11 \)[/tex] is:
[tex]\[ x = \ln(2) - 1 \][/tex]
Comparing this solution with the provided choices:
1. [tex]\( x = \ln 2 - 1 \)[/tex]
2. [tex]\( x = \ln 2 + 1 \)[/tex]
3. [tex]\( x = \frac{1}{e} \)[/tex]
4. [tex]\( x = \frac{e + 2}{e} \)[/tex]
We see that the correct match is:
[tex]\[ x = \ln 2 - 1 \][/tex]
