IDNLearn.com is designed to help you find reliable answers to any question you have. Our community is here to provide the comprehensive and accurate answers you need to make informed decisions.
Sagot :
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]
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]
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.