Pregunta
Use properties of logarithms to evaluate the expression. \( \ln e^{-4}-\ln e^{-\frac{3}{4}} \)
Ask by Pope Schmidt. in the United States
Jan 24,2025
Solución de tutoría real
Respuesta verificada por el tutor
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The expression simplifies to \( -3.25 \).
Solución
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 \( -3.25 \).
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Simplifique esta solución Beyond the Answer
First, let's simplify each term using the property that \( \ln e^x = x \). So, \( \ln e^{-4} = -4 \) and \( \ln e^{-\frac{3}{4}} = -\frac{3}{4} \). Now, substituting these values into the expression, we get: \[ \ln e^{-4}-\ln e^{-\frac{3}{4}} = -4 - \left(-\frac{3}{4}\right) = -4 + \frac{3}{4} \] Next, we can convert -4 into a fraction to combine the two terms. Since -4 can be written as \(-\frac{16}{4}\), the expression becomes: \[ -\frac{16}{4} + \frac{3}{4} = -\frac{16 - 3}{4} = -\frac{13}{4} \] Thus, the final evaluation of the expression is: \[ \ln e^{-4}-\ln e^{-\frac{3}{4}} = -\frac{13}{4} \]
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