91Ó°ÊÓ

Logarithmic expansion involves using the properties of logarithms to rewrite complex logarithmic expressions in a simpler form. This is often done to make calculations easier or to better understand the structure of the expression.

In the given exercise, we're asked to expand the expression \(\ln \sqrt{z}\). First, we recognize that a square root can be written as an exponent: \(\sqrt{z} = z^{1/2}\). Using this property helps transform the original problem into something we can further work with.

• Expression: \(\ln \sqrt{z}\)
• Re-written as: \(\ln(z^{1/2})\)

Understanding this rewrite is essential in manipulating the expression to reveal its expanded form, as the square root becomes a power, leading us to utilize further logarithmic properties.

The power rule of logarithms is a key tool for expanding logarithmic expressions. It is based on the principle that an exponent inside a logarithm can be moved in front as a multiplier.

Mathematically, this is expressed as \(\ln{a^b} = b \cdot \ln{a}\). This rule is powerful because it allows us to take a complex expression and simplify it by separating the power from the logarithm itself.
For our exercise, applying the power rule to \(\ln(z^{1/2})\), we shift the \(1/2\) in front of the logarithm, transforming the expression into:

• \(\ln(z^{1/2}) = \frac{1}{2} \cdot \ln z\)

By doing this, the logarithm is expanded into a simpler and more manageable form. The multiplier \(1/2\) provides a clear indication of how the original term relates to \(\ln z\). This makes further manipulation and interpretation straightforward.

When working with logarithms, it's crucial to understand the restrictions on variables. Logarithms are undefined for non-positive numbers, meaning the variables must be positive to ensure the expression is valid.

In our exercise, the assumption is that all variables are positive, which means we can safely expand and simplify without encountering undefined behavior.
Here's why it matters:

• Logarithm of zero or a negative number is undefined.
• Ensuring \(z > 0\) allows \(\ln z\) and by extension \(\ln(z^{1/2})\) to be valid expressions.

This restriction is not merely a formality; it ensures that our mathematical operations hold true across all steps. Misunderstanding this can lead to errors in both reasoning and calculation.