Question

Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
ln left parenthesis StartFraction e Superscript 7 Over 11 EndFraction right parenthesis
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Part 1
ln left parenthesis StartFraction e Superscript 7 Over 11 EndFraction right parenthesis
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Answer 1

To expand the expression \(\ln\left(\frac{e^7}{11}\right)\) using properties of logarithms, we can apply the following properties:

1. \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
2. \(\ln(a^b) = b \cdot \ln(a)\)
3. \(\ln(e) = 1\)

Using these properties, we can rewrite \(\ln\left(\frac{e^7}{11}\right)\) as follows:

\[
\ln\left(\frac{e^7}{11}\right) = \ln(e^7) - \ln(11)
\]

Next, we use the property of logarithms for \(e^7\):

\[
\ln(e^7) = 7 \cdot \ln(e) = 7 \cdot 1 = 7
\]

Now, we can combine this back into the expression:

\[
\ln\left(\frac{e^7}{11}\right) = 7 - \ln(11)
\]

So, the expanded expression is:

\[
\ln\left(\frac{e^7}{11}\right) = 7 - \ln(11)
\]

This is as far as we can expand it without a calculator.