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What is the solution to [tex]4 + 5e^{x+2} = 11[/tex]?

A. [tex]x = \ln \frac{7}{5} - 2[/tex]
B. [tex]x = \ln \frac{7}{5} + 2[/tex]
C. [tex]x = \ln 35 - 2[/tex]
D. [tex]x = \ln 35 + 2[/tex]


Sagot :

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]