Problem 18 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \ln \sqrt{z} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \ln(z) \)

Step 1: Represent the Square Root Using Exponents

The square root of a number can be expressed as an exponent. Rewrite the expression \( \sqrt{z} \) as \( z^{1/2} \). The expression becomes \( \ln(z^{1/2}) \).

Step 2: Apply the Power Rule for Logarithms

According to the Power Rule of Logarithms, \( \ln(a^b) = b \cdot \ln(a) \). Apply this to the expression \( \ln(z^{1/2}) \). This results in \( \frac{1}{2} \cdot \ln(z) \).

Step 3: Final Simplification

The expression is now fully expanded. The expanded expression is \( \frac{1}{2} \ln(z) \).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are powerful mathematical tools that allow us to transform multiplication into addition and exponentiation into multiplication. They are inversely related to exponential expressions. In the expression \( \ln(\sqrt{z}) \), "ln" denotes the natural logarithm, which has a base of \( e \), approximately equal to 2.718. The function \( \ln(x) \) answers the question: "To what power must \( e \) be raised, to yield \( x \)?".
Understanding logarithmic expressions involves recognizing these transformations. Logarithms simplify complex calculations involving multiplication and division into simpler operations of addition and subtraction. Here, the goal is to expand \( \ln(\sqrt{z}) \), making it easier to work with.
This example shows how logarithmic expressions help manage complex operations. Through the rules we'll discuss, you can see how simplifying logarithms can lead to more efficient problem-solving in calculus and algebra.

Power Rule for Logarithms

The Power Rule for logarithms makes dealing with exponents straightforward. It states: \( \ln(a^b) = b \cdot \ln(a) \). Think of it as bringing the exponent out in front of the logarithm. This allows you to manage the expression easily, especially when the exponent is a fraction or any other real number.
For the given expression \( \ln(z^{1/2}) \), use the Power Rule to extract the exponent \( \frac{1}{2} \) and place it in front of the logarithm, simplifying computation. Applying the Power Rule gives us \( \frac{1}{2} \cdot \ln(z) \).
Effectively using the Power Rule can drastically change how you work with logarithmic statements. It simplifies complicated expressions, making calculations faster and less error-prone.

• Helps in breaking down complex exponential expressions.
• Simplifies the expression by removing the exponent inside the log.

Expanding Logarithms

Expanding logarithms is about breaking a complex logarithmic expression into simpler parts without altering its value. In the process, you use several properties, including the Power Rule, the Product Rule, and the Quotient Rule.
Expanding helps particularly in integral calculus and in solving differential equations, where simplified expressions make computations tractable. To expand the expression \( \ln(\sqrt{z}) \), first recognize \( \sqrt{z} \) as \( z^{1/2} \).
Then apply the Power Rule to expand it to \( \frac{1}{2} \ln(z) \), which is simpler to work with. Thus, you've taken a potentially hard-to-evaluate expression and rewritten it to be more straightforward.
This method:

• Makes logarithmic expressions easy to interpret and apply.
• Breaks down exponential terms outside the logarithmic function.

Expanding logarithms not only aids in making calculations less daunting but also allows deeper insights into the behaviour of expressions involved.