amelialang07
Answered

What is the value of \( x \) in the equation below?

[tex]\[ 1 + 2e^{x+1} = 9 \][/tex]

A. \( x = \log 4 - 1 \)
B. \( x = \log 4 \)
C. \( x = \ln 4 - 1 \)
D. [tex]\( x = \ln 4 \)[/tex]



Answer :

GinnyAnswer
To solve the equation \(1 + 2 e^{x+1} = 9\) for \(x\), let's work through it step by step.

1. Start with the original equation:
[tex]\[ 1 + 2 e^{x+1} = 9 \][/tex]

2. Isolate the exponential term:
[tex]\[ 2 e^{x+1} = 9 - 1 \][/tex]
Simplify the right-hand side:
[tex]\[ 2 e^{x+1} = 8 \][/tex]

3. Divide both sides by 2 to solve for \(e^{x+1}\):
[tex]\[ e^{x+1} = \frac{8}{2} \][/tex]
Simplify the fraction:
[tex]\[ e^{x+1} = 4 \][/tex]

4. Take the natural logarithm (ln) on both sides to solve for \(x+1\):
[tex]\[ \ln(e^{x+1}) = \ln(4) \][/tex]

5. Use the property of logarithms \(\ln(e^y) = y\):
[tex]\[ x + 1 = \ln(4) \][/tex]

6. Solve for \(x\) by subtracting 1 from both sides:
[tex]\[ x = \ln(4) - 1 \][/tex]

Given the choices:
- \(x = \log 4 - 1\)
- \(x = \log 4\)
- \(x = \ln 4 - 1\)
- \(x = \ln 4\)

The value that matches is:

[tex]\[ x = \ln 4 - 1 \][/tex]

Using the numerical result, we can confirm that this is approximately:
[tex]\[ x = 0.3862943611198906 \][/tex]

Therefore, the correct option is:
[tex]\[ x = \ln 4 - 1 \][/tex]

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