Problem 24 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \ln \sqrt{z} $$

Step-by-Step Solution

Answer

The expanded form is \( \frac{1}{2} \ln(z) \).

Step 1: Convert the Square Root to Exponent

The square root of a number can be written as an exponent. Therefore, \( \sqrt{z} \) is rewritten as \( z^{1/2} \).

Step 2: Apply the Power Rule for Logarithms

The power rule for logarithms states that \( \ln(a^b) = b \cdot \ln(a) \). Applying this rule, \( \ln(z^{1/2}) = \frac{1}{2} \ln(z) \).

Key Concepts

Logarithmic Expansion

Logarithmic expansion involves transforming a complex logarithmic expression into a simpler or more detailed form, using various properties of logarithms. These properties make it easier to perform calculations and manipulate expressions efficiently. When expanding logarithmic expressions, you are essentially breaking them down into more manageable parts.

With the given exercise, we focus on transforming a single expression. Initially, the expression given is the natural logarithm of a square root, which already hints at specific laws that could be applied. By applying these rules, we aim to create an expanded expression that often makes it easier to differentiate expressions or solve equations involving logarithms. Understanding how to expand logarithms using their laws is critical in different areas of mathematics, such as calculus and algebra. It forms a foundational skill that aids not just in theoretical discussions, but also in practical problems.

Square Root as Exponent

The concept of a square root can be seamlessly expressed as an exponent. This transformation is a crucial step in simplifying various mathematical expressions, particularly those involving logarithms. By expressing a square root in terms of an exponent, you can apply additional rules and properties conveniently.

A square root, represented with the radical symbol, can be rewritten using fractional exponents. For example, if you have \( \sqrt{z} \), it is equivalent to \( z^{1/2} \). The index of the root becomes the denominator of the fractional exponent. This conversion simplifies operations, especially when applying logarithm rules.

• Square root of any number \(x\): \( \sqrt{x} = x^{1/2} \)
• Cube root of any number \(x\): \( \sqrt[3]{x} = x^{1/3} \)

Being comfortable with this conversion helps tremendously in dealing with and simplifying expressions involving logarithms.

Power Rule for Logarithms

The power rule for logarithms is a handy tool that allows you to deal with logarithmic expressions that involve exponents. This rule states that when you have a logarithm of a number raised to an exponent, you can bring the exponent down as a coefficient. Formally, the rule is expressed as:

\[ \ln(a^b) = b \cdot \ln(a)\]

In the given exercise, after converting the square root into a fractional exponent, we were able to straightforwardly apply this power rule. For example, \( \ln(z^{1/2}) \) is simplified to \( \frac{1}{2} \ln(z) \). This manipulation makes it significantly easier to handle the expression, especially in integration and differentiation in calculus.

The power rule is not limited to natural logarithms; it applies to logs of any base. Understanding this rule is essential for effectively managing expressions with exponents within logarithms. It streamlines complex calculations and opens up a pathway to more advanced problem solving in different mathematical contexts where logarithms are used.