To solve the equation \(1 + 2 e^{x+1} = 9\) for \(x\), let's work through it step by step.
1. Start with the original equation:
[tex]\[
1 + 2 e^{x+1} = 9
\][/tex]
2. Isolate the exponential term:
[tex]\[
2 e^{x+1} = 9 - 1
\][/tex]
Simplify the right-hand side:
[tex]\[
2 e^{x+1} = 8
\][/tex]
3. Divide both sides by 2 to solve for \(e^{x+1}\):
[tex]\[
e^{x+1} = \frac{8}{2}
\][/tex]
Simplify the fraction:
[tex]\[
e^{x+1} = 4
\][/tex]
4. Take the natural logarithm (ln) on both sides to solve for \(x+1\):
[tex]\[
\ln(e^{x+1}) = \ln(4)
\][/tex]
5. Use the property of logarithms \(\ln(e^y) = y\):
[tex]\[
x + 1 = \ln(4)
\][/tex]
6. Solve for \(x\) by subtracting 1 from both sides:
[tex]\[
x = \ln(4) - 1
\][/tex]
Given the choices:
- \(x = \log 4 - 1\)
- \(x = \log 4\)
- \(x = \ln 4 - 1\)
- \(x = \ln 4\)
The value that matches is:
[tex]\[
x = \ln 4 - 1
\][/tex]
Using the numerical result, we can confirm that this is approximately:
[tex]\[
x = 0.3862943611198906
\][/tex]
Therefore, the correct option is:
[tex]\[
x = \ln 4 - 1
\][/tex]
