Question
116. Solve: \( 9 e^{3 x}-4=32 \). Find the solution set and then use a calculator to obtain a decimal approximation to two decimal places for the solution.
Ask by Henry Goodwin. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is \( x = \dfrac{\ln 4}{3} \), which is approximately 0.46.
Solution
To solve the equation \( 9 e^{3x} - 4 = 32 \), follow these steps:
1. **Isolate the exponential term:**
\[
9 e^{3x} - 4 = 32 \\
9 e^{3x} = 32 + 4 \\
9 e^{3x} = 36 \\
e^{3x} = \frac{36}{9} \\
e^{3x} = 4
\]
2. **Take the natural logarithm of both sides to solve for \( x \):**
\[
\ln(e^{3x}) = \ln(4) \\
3x = \ln(4) \\
x = \frac{\ln(4)}{3}
\]
3. **Solution Set:**
\[
\left\{ \frac{\ln(4)}{3} \right\}
\]
4. **Decimal Approximation:**
\[
\ln(4) \approx 1.38629 \\
x \approx \frac{1.38629}{3} \approx 0.4621
\]
Rounded to two decimal places:
\[
x \approx 0.46
\]
**Final Answer:**
The solution is \( x = \dfrac{\ln 4}{3} \). In decimal form, \( x \approx 0.46 \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To solve the equation \( 9 e^{3 x} - 4 = 32 \), we start by rearranging it: 1. Add 4 to both sides: \[ 9 e^{3 x} = 36 \] 2. Divide both sides by 9: \[ e^{3 x} = 4 \] 3. Now, to get rid of the exponential, take the natural logarithm of both sides: \[ 3 x = \ln(4) \] 4. Divide by 3: \[ x = \frac{\ln(4)}{3} \] Now, using a calculator, we can find \( \ln(4) \): \[ \ln(4) \approx 1.3863 \] Thus, \[ x \approx \frac{1.3863}{3} \approx 0.4621 \] Rounding to two decimal places, the decimal approximation for the solution is: \[ x \approx 0.46 \] The solution set is: \[ \left\{ \frac{\ln(4)}{3} \right\} \] with the decimal approximation \( x \approx 0.46 \).
