Let's solve the given equation step-by-step to find the value of [tex]\( x \)[/tex]:
Given equation:
[tex]\[ 1 + 2 e^{x+1} = 9 \][/tex]
1. Start by isolating the exponential term. Subtract 1 from both sides:
[tex]\[ 2 e^{x+1} = 8 \][/tex]
2. Divide both sides by 2 to further isolate the exponential component:
[tex]\[ e^{x+1} = 4 \][/tex]
3. To solve for [tex]\( x \)[/tex], take the natural logarithm (ln) of both sides. The natural logarithm of an exponential function simplifies nicely:
[tex]\[ \ln(e^{x+1}) = \ln(4) \][/tex]
Using the property of logarithms that [tex]\(\ln(e^y) = y\)[/tex]:
[tex]\[ x + 1 = \ln(4) \][/tex]
4. Finally, solve for [tex]\( x \)[/tex] by subtracting 1 from both sides:
[tex]\[ x = \ln(4) - 1 \][/tex]
Therefore, the correct value of [tex]\( x \)[/tex] is:
[tex]\[ x = \ln(4) - 1 \][/tex]
Among the provided options, the correct option is:
[tex]\[ x = \ln 4 - 1 \][/tex]
The numerical evaluation of this expression is approximately [tex]\( 0.3862943611198906 \)[/tex], which confirms the solution.
Thus, the value of [tex]\( x \)[/tex] in the equation is [tex]\( \ln 4 - 1 \)[/tex], corresponding to the approximate value of [tex]\( 0.3862943611198906 \)[/tex].
