Question
Use the ALEKS calculator to evaluate each expression. Round your answers to the nearest thousandth. Do not round any intermediate computations. \[ \ln \frac{3}{4}=\square \] \( \ln 32.5=\square \)
Ask by Young Harris. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\ln \frac{3}{4} = -0.288
\]
\[
\ln 32.5 = 3.481
\]
Solution
Sure, let's evaluate each expression using precise calculations and then round the results to the nearest thousandth.
1. **\(\ln \frac{3}{4}\)**
\[
\ln \left(\frac{3}{4}\right) = \ln(0.75) \approx -0.2876820724517809
\]
Rounded to the nearest thousandth:
\[
\ln \frac{3}{4} = -0.288
\]
2. **\(\ln 32.5\)**
\[
\ln(32.5) \approx 3.481240092319817
\]
Rounded to the nearest thousandth:
\[
\ln 32.5 = 3.481
\]
**Final Answers:**
\[
\ln \frac{3}{4} = -0.288
\]
\[
\ln 32.5 = 3.481
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To evaluate the expression \(\ln \frac{3}{4}\), use the property of logarithms that states \(\ln \frac{a}{b} = \ln a - \ln b\). Thus, you would calculate \(\ln 3 \approx 1.0986\) and \(\ln 4 \approx 1.3863\). When you subtract these values, \(\ln \frac{3}{4} \approx 1.0986 - 1.3863 = -0.2877\). Rounding to the nearest thousandth yields \(-0.288\). For the expression \(\ln 32.5\), simply find the natural logarithm of 32.5, which is approximately \(3.471\). Rounding this to the nearest thousandth gives \(\ln 32.5 \approx 3.471\). So your answers are: \[ \ln \frac{3}{4} \approx -0.288 \] \[ \ln 32.5 \approx 3.471 \]
