Questions: Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate ln(e^4/11)

Solution Steps

To expand the logarithmic expression \(\ln \left(\frac{e^{4}}{11}\right)\), we can use the properties of logarithms. Specifically, we will use the quotient rule for logarithms, which states that \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\). Additionally, we will use the power rule, which states that \(\ln(a^b) = b \cdot \ln(a)\).

1. Apply the quotient rule: \(\ln \left(\frac{e^{4}}{11}\right) = \ln(e^4) - \ln(11)\).
2. Apply the power rule to \(\ln(e^4)\): \(\ln(e^4) = 4 \cdot \ln(e)\).
3. Since \(\ln(e) = 1\), simplify the expression.

We start with the expression: \[ \ln \left(\frac{e^{4}}{11}\right) \] Using the quotient rule for logarithms, we can rewrite this as: \[ \ln(e^4) - \ln(11) \]

Next, we apply the power rule to \(\ln(e^4)\): \[ \ln(e^4) = 4 \cdot \ln(e) \] Since \(\ln(e) = 1\), we have: \[ \ln(e^4) = 4 \cdot 1 = 4 \]

Now we can substitute back into our expression: \[ \ln \left(\frac{e^{4}}{11}\right) = 4 - \ln(11) \] Using the numerical value of \(\ln(11) \approx 2.3979\), we find: \[ \ln \left(\frac{e^{4}}{11}\right) \approx 4 - 2.3979 = 1.6021 \]

Final Answer