Question
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. (If there is no solution, enter NO SOLUTION. \[ 2+7 \ln (x)=5 \] \( x=\square \) Watch n .
Ask by Nichols Grant. in the United States
Dec 07,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x \approx 1.535 \)
Solution
Solve the equation \( 2+7 \ln (x)=5 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(2+7\ln{\left(x\right)}=5\)
- step1: Find the domain:
\(2+7\ln{\left(x\right)}=5,x>0\)
- step2: Move the expression to the left side:
\(2+7\ln{\left(x\right)}-5=0\)
- step3: Subtract the numbers:
\(-3+7\ln{\left(x\right)}=0\)
- step4: Rewrite the expression:
\(7\ln{\left(x\right)}=3\)
- step5: Divide both sides:
\(\frac{7\ln{\left(x\right)}}{7}=\frac{3}{7}\)
- step6: Divide the numbers:
\(\ln{\left(x\right)}=\frac{3}{7}\)
- step7: Convert the logarithm into exponential form:
\(x=e^{\frac{3}{7}}\)
- step8: Transform the expression:
\(x=\sqrt[7]{e^{3}}\)
- step9: Check if the solution is in the defined range:
\(x=\sqrt[7]{e^{3}},x>0\)
- step10: Find the intersection:
\(x=\sqrt[7]{e^{3}}\)
The solution to the logarithmic equation \(2+7 \ln (x)=5\) is \(x \approx 1.535\).
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Simplify this solution Mind Expander
To solve the equation \( 2 + 7 \ln(x) = 5 \), first, isolate the logarithmic term. Subtract 2 from both sides: \[ 7 \ln(x) = 3 \] Next, divide both sides by 7: \[ \ln(x) = \frac{3}{7} \] Now, exponentiate both sides to eliminate the logarithm: \[ x = e^{\frac{3}{7}} \] Using a calculator, approximate \( e^{\frac{3}{7}} \): \( x \approx 1. quitté jackson 150 \) So, the final result is: \( x \approx 1.512 \)
