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 4 StartRoot x EndRoot
Question content area bottom
Part 1
ln RootIndex 4 StartRoot x EndRootequals
  
enter your response here

Answer 1

To expand the expression \( \ln \left( \sqrt[4]{x} \right) \) using properties of logarithms, we will apply a couple of logarithmic properties:

1. The property of logarithms that states \( \ln(a^b) = b \cdot \ln(a) \).
2. The property of logarithms that allows us to express roots as exponents, \( \sqrt[n]{a} = a^{1/n} \).

First, we can express \( \sqrt[4]{x} \) as \( x^{1/4} \). Therefore, we rewrite the logarithm:

\[
\ln \left( \sqrt[4]{x} \right) = \ln \left( x^{1/4} \right)
\]

Now, applying the first property, we can expand this further:

\[
\ln \left( x^{1/4} \right) = \frac{1}{4} \ln(x)
\]

Thus, the expanded form of \( \ln \left( \sqrt[4]{x} \right) \) is:

\[
\ln \left( \sqrt[4]{x} \right) = \frac{1}{4} \ln(x)
\]

So, the final answer is:

\[
\frac{1}{4} \ln(x)
\]