To solve the equation \(4 + 5e^{x + 2} = 11\), we'll follow these steps:
### Step 1: Isolate the exponential term
First, we need to isolate the term containing the exponential function:
[tex]\[ 4 + 5e^{x + 2} = 11 \][/tex]
Subtract 4 from both sides:
[tex]\[ 5e^{x + 2} = 7 \][/tex]
### Step 2: Isolate the exponential
Next, let's isolate \( e^{x + 2} \) by dividing both sides by 5:
[tex]\[ e^{x + 2} = \frac{7}{5} \][/tex]
### Step 3: Apply the natural logarithm
To solve for \(x\), we'll take the natural logarithm of both sides. The natural logarithm (\(\ln\)) will help us bring down the exponent:
[tex]\[ \ln(e^{x + 2}) = \ln\left(\frac{7}{5}\right) \][/tex]
Using the property of logarithms \(\ln(e^a) = a\), we get:
[tex]\[ x + 2 = \ln\left(\frac{7}{5}\right) \][/tex]
### Step 4: Solve for \(x\)
Finally, solve for \(x\) by subtracting 2 from both sides:
[tex]\[ x = \ln\left(\frac{7}{5}\right) - 2 \][/tex]
### Verification with Given Choices
We need to verify which of the given choices matches our solution:
- \( x = \ln\left(\frac{7}{5}\right) - 2 \)
- \( x = \ln\left(\frac{7}{5}\right) + 2 \)
- \( x = \ln(35) - 2 \)
- \( x = \ln(35) + 2 \)
From the step-by-step solution, the correct answer is:
[tex]\[ x = \ln\left(\frac{7}{5}\right) - 2 \][/tex]
So, the solution to the equation \(4 + 5e^{x + 2} = 11\) is:
[tex]\[ x = \ln\left(\frac{7}{5}\right) - 2 \][/tex]
This matches the choice:
[tex]\[ x = \ln\left(\frac{7}{5}\right) - 2 \][/tex]
