To solve the equation [tex]\( 4 + 5e^{x+2} = 11 \)[/tex], follow these steps:
1. Isolate the exponential expression:
Start by subtracting 4 from both sides of the equation to isolate the term involving the exponential function:
[tex]\[
4 + 5e^{x+2} = 11 \implies 5e^{x+2} = 11 - 4 \implies 5e^{x+2} = 7
\][/tex]
2. Divide by 5:
Divide both sides of the equation by 5 to further isolate the exponential function:
[tex]\[
e^{x+2} = \frac{7}{5}
\][/tex]
3. Take the natural logarithm of both sides:
To solve for [tex]\( x \)[/tex], take the natural logarithm (ln) of both sides. Recall that [tex]\( \ln(e^y) = y \)[/tex]:
[tex]\[
\ln(e^{x+2}) = \ln\left(\frac{7}{5}\right)
\][/tex]
4. Simplify using properties of logarithms:
On the left side, use the property that [tex]\( \ln(e^y) = y \)[/tex]:
[tex]\[
x + 2 = \ln\left(\frac{7}{5}\right)
\][/tex]
5. Isolate [tex]\( x \)[/tex]:
Subtract 2 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \ln\left(\frac{7}{5}\right) - 2
\][/tex]
Now, let's compare this form to the given options:
- [tex]\( x = \ln\left(\frac{7}{5}\right) - 2 \)[/tex]
- [tex]\( x = \ln\left(\frac{7}{5}\right) + 2 \)[/tex]
- [tex]\( x = \ln 35 - 2 \)[/tex]
- [tex]\( x = \ln 35 + 2 \)[/tex]
The correct option based on our step-by-step solution is:
[tex]\[ x = \ln\left(\frac{7}{5}\right) - 2 \][/tex]
The numerical value for this option can be evaluated and is approximately [tex]\(-1.66353\)[/tex].
So, the solution to the equation is:
[tex]\[ x = \ln\left(\frac{7}{5}\right) - 2 \][/tex]
and this corresponds to option [tex]\(\boxed{x = \ln\left(\frac{7}{5}\right) - 2}\)[/tex].
