Question
Use properties of logarithms to evaluate the expression. \( \ln e^{-4}-\ln e^{-\frac{3}{4}} \)
Ask by Martin Cox. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( -3.25 \).
Solution
Calculate or simplify the expression \( \ln(e^{-4})-\ln(e^{-\frac{3}{4}}) \).
Calculate the value by following steps:
- step0: Calculate:
\(\ln{\left(e^{-4}\right)}-\ln{\left(e^{-\frac{3}{4}}\right)}\)
- step1: Simplify the expression:
\(-4-\ln{\left(e^{-\frac{3}{4}}\right)}\)
- step2: Simplify the expression:
\(-4-\left(-\frac{3}{4}\right)\)
- step3: Remove the parentheses:
\(-4+\frac{3}{4}\)
- step4: Reduce fractions to a common denominator:
\(-\frac{4\times 4}{4}+\frac{3}{4}\)
- step5: Transform the expression:
\(\frac{-4\times 4+3}{4}\)
- step6: Multiply the numbers:
\(\frac{-16+3}{4}\)
- step7: Add the numbers:
\(\frac{-13}{4}\)
- step8: Rewrite the fraction:
\(-\frac{13}{4}\)
The expression \( \ln e^{-4}-\ln e^{-\frac{3}{4}} \) simplifies to \( -\frac{13}{4} \) or \( -3\frac{1}{4} \) or \( -3.25 \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To evaluate the expression \( \ln e^{-4}-\ln e^{-\frac{3}{4}} \), we can use the properties of logarithms. First, we apply the property that states \( \ln e^x = x \). This simplifies our expression to: \[ -4 - \left(-\frac{3}{4}\right) = -4 + \frac{3}{4} \] To combine the terms, we can rewrite \(-4\) as \(-\frac{16}{4}\). Thus, we have: \[ -\frac{16}{4} + \frac{3}{4} = -\frac{16 - 3}{4} = -\frac{13}{4} \] So, the final evaluation of the expression \( \ln e^{-4}-\ln e^{-\frac{3}{4}} \) is: \[ -\frac{13}{4} \]
