Question
Solve for \( x: 2 e^{x+1}-6=10 \) \( \begin{array}{lll}1.079 & \text { b. } & 3.079 \\ 2,979.958 & \text { d. } & 2,981.958\end{array} \)
Ask by Ray Johnson. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is approximately 1.079.
Solution
Solve the equation \( 2e^{x+1}-6=10 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(2e^{x+1}-6=10\)
- step1: Move the expression to the left side:
\(2e^{x+1}-6-10=0\)
- step2: Subtract the numbers:
\(2e^{x+1}-16=0\)
- step3: Rewrite the expression:
\(2e^{x+1}=16\)
- step4: Divide both sides:
\(\frac{2e^{x+1}}{2}=\frac{16}{2}\)
- step5: Divide the numbers:
\(e^{x+1}=8\)
- step6: Take the logarithm of both sides:
\(\ln{\left(e^{x+1}\right)}=\ln{\left(8\right)}\)
- step7: Evaluate the logarithm:
\(x+1=\ln{\left(8\right)}\)
- step8: Move the constant to the right side:
\(x=\ln{\left(8\right)}-1\)
- step9: Simplify:
\(x=3\ln{\left(2\right)}-1\)
The solution to the equation \(2e^{x+1}-6=10\) is \(x=3\ln{(2)}-1\).
Now, let's calculate the value of \(3\ln{(2)}-1\) to determine which option is correct.
Calculate the value by following steps:
- step0: Calculate:
\(3\ln{\left(2\right)}-1\)
The value of \(3\ln{(2)}-1\) is approximately 1.079442.
Therefore, the correct option is 1.079.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To solve the equation \( 2 e^{x+1}-6=10 \), we first isolate the exponential term: 1. Add 6 to both sides: \[ 2 e^{x+1} = 16 \] 2. Divide both sides by 2: \[ e^{x+1} = 8 \] 3. Now, take the natural logarithm of both sides: \[ x + 1 = \ln(8) \] 4. Finally, solve for \( x \): \[ x = \ln(8) - 1 \] Using the fact that \( \ln(8) = \ln(2^3) = 3 \ln(2) \), we can compute \( x \). Calculating \( \ln(2) \approx 0.6931 \), we get: \[ \ln(8) \approx 3 \times 0.6931 \approx 2.0793 \] Thus, \[ x \approx 2.0793 - 1 \approx 1.0793 \] The closest answer is \( 1.079 \).
