Question
Solve for \( x \) in the following equation. Express your answer to three significant figures. \( 0.912=\ln \left(x^{3.00}\right) \)
Ask by Simpson Ward. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x \approx 1.36 \)
Solution
To solve the equation \( 0.912 = \ln \left(x^{3.00}\right) \) for \( x \), follow these steps:
1. **Simplify the Equation:**
\[
0.912 = \ln(x^3)
\]
Using the logarithm power rule \( \ln(x^n) = n \ln(x) \):
\[
0.912 = 3 \ln(x)
\]
2. **Isolate \( \ln(x) \):**
\[
\ln(x) = \frac{0.912}{3} = 0.304
\]
3. **Exponentiate Both Sides to Solve for \( x \):**
\[
x = e^{0.304}
\]
4. **Calculate \( e^{0.304} \):**
- Using a calculator or series expansion:
\[
e^{0.304} \approx 1.355
\]
5. **Express to Three Significant Figures:**
\[
x \approx 1.36
\]
**Final Answer:**
\( x \approx 1.36 \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To solve for \( x \) in the equation \( 0.912=\ln \left(x^{3.00}\right) \), we start by using the properties of logarithms. We can rewrite the equation as: \[ \ln \left(x^{3.00}\right) = 3 \ln(x) \] Thus, we have: \[ 0.912 = 3 \ln(x) \] Next, we divide both sides by 3 to isolate \( \ln(x) \): \[ \ln(x) = \frac{0.912}{3} \approx 0.304 \] Now, we exponentiate both sides to solve for \( x \): \[ x = e^{0.304} \] Calculating \( e^{0.304} \) gives approximately \( 1.355 \). Rounding to three significant figures, we find: \[ x \approx 1.36 \] Thus, the solution for \( x \) is: \[ \boxed{1.36} \]
