Let's solve the equation [tex]\( 4 + 5e^{x+2} = 11 \)[/tex] step-by-step.
1. Isolate the exponential expression:
[tex]\[
4 + 5e^{x+2} = 11
\][/tex]
Subtract 4 from both sides to isolate the term with the exponential function:
[tex]\[
5e^{x+2} = 7
\][/tex]
2. Solve for the exponent:
Divide both sides by 5 to isolate the exponential expression:
[tex]\[
e^{x+2} = \frac{7}{5}
\][/tex]
3. Apply the natural logarithm:
To solve for [tex]\( x \)[/tex], take the natural logarithm (ln) on both sides:
[tex]\[
\ln(e^{x+2}) = \ln \left( \frac{7}{5} \right)
\][/tex]
4. Simplify using properties of logarithms:
The natural logarithm and the exponential function are inverse functions, so:
[tex]\[
x + 2 = \ln \left( \frac{7}{5} \right)
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Subtract 2 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \ln \left( \frac{7}{5} \right) - 2
\][/tex]
Checking the given options:
- [tex]\( x = \ln \frac{7}{5} - 2 \)[/tex]
- [tex]\( x = \ln \frac{7}{5} + 2 \)[/tex]
- [tex]\( x = \ln 35 - 2 \)[/tex]
- [tex]\( x = \ln 35 + 2 \)[/tex]
The correct solution to the equation [tex]\( 4 + 5e^{x+2} = 11 \)[/tex] is:
[tex]\[
x = \ln \frac{7}{5} - 2
\][/tex]
