Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
ln RootIndex 11 StartRoot x EndRoot
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Part 1
ln RootIndex 11 StartRoot x EndRootequals
  
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Answer 1

To expand the logarithmic expression \(\ln \sqrt[11]{\sqrt{x}}\), we can first rewrite the expression in a more manageable form and then apply the properties of logarithms.

1. **Rewrite the expression**:
\[
\sqrt[11]{\sqrt{x}} = (\sqrt{x})^{1/11} = (x^{1/2})^{1/11} = x^{1/22}
\]

2. **Apply the logarithm**:
\[
\ln(x^{1/22})
\]

3. **Use the property of logarithms**: The power rule states that \(\ln(a^b) = b \cdot \ln(a)\).
\[
\ln(x^{1/22}) = \frac{1}{22} \ln(x)
\]

Thus, the expanded form of the logarithmic expression is:
\[
\ln \sqrt[11]{\sqrt{x}} = \frac{1}{22} \ln(x)
\]

This is the final answer.