Question
\( ? \) QUESTION Solve for \( x \) \( 3 \ln (x-6)=15 \)
Ask by Patel Hart. in the United States
Mar 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = e^{5} + 6 \)
Solution
Solve the equation \( 3\ln(x-6)=15 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(3\ln{\left(x-6\right)}=15\)
- step1: Find the domain:
\(3\ln{\left(x-6\right)}=15,x>6\)
- step2: Divide both sides:
\(\frac{3\ln{\left(x-6\right)}}{3}=\frac{15}{3}\)
- step3: Divide the numbers:
\(\ln{\left(x-6\right)}=5\)
- step4: Convert the logarithm into exponential form:
\(x-6=e^{5}\)
- step5: Move the constant to the right side:
\(x=e^{5}+6\)
- step6: Check if the solution is in the defined range:
\(x=e^{5}+6,x>6\)
- step7: Find the intersection:
\(x=e^{5}+6\)
The solution to the equation \(3\ln(x-6)=15\) is \(x=e^{5}+6\).
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Simplify this solution Bonus Knowledge
First, let's isolate the natural logarithm by dividing both sides by 3: \(\ln(x-6) = 5\). Now, to eliminate the logarithm, we exponentiate both sides using the base \( e \): \(x-6 = e^5\). Therefore, \(x = e^5 + 6\). You can compute \( e^5 \) to get a numerical value, approximately \( 148.41 \), leading to \(x \approx 154.41\). To verify your solution, plug \(x\) back into the original equation. If the left-hand side equals the right-hand side, you’ve successfully arrived at the correct solution!
