Problem 20 - Calculus of a Single Variable

Question

In Exercises 19–28, use the properties of logarithms to expand the logarithmic expression. $$ \ln \sqrt{x^{5}} $$

Step-by-Step Solution

Answer

The expanded form of \( \ln \sqrt{x^{5}} \) is \( \frac{5}{2} \ln x \).

Step 1: Convert square root into a power with exponent

Rewrite \( \sqrt{x^{5}} \) as \( x^{5/2} \). So, the original expression can be rewritten as \( \ln x^{5/2} \)

Step 2: Using the Power Rule of Logarithms

Apply the power rule of logarithms that states \( \ln (m^n) = n \ln m \). This transforms \( \ln x^{5/2} \) into \( \frac{5}{2} \ln x \).

Key Concepts

Expanding Logarithmic Expressions

When expanding logarithmic expressions, our goal is to take a complex log expression and break it down into a simpler form using the properties of logarithms. Let's consider the expression \( \ln \sqrt{x^{5}} \). At first glance, the presence of both a square root and exponent might seem a bit intimidating.

However, one key approach is recognizing that the square root can be translated into an exponent. In this case, \( \sqrt{x^{5}} \) can be rewritten as \( x^{5/2} \). This new form helps simplify calculating the logarithm. By applying these transformations, you break down multi-layered equations into more manageable pieces.

Expanding and manipulating logarithms isn't just a tool for simplifying expressions. It also aids in solving more complex equations and understanding behavior in exponential functions. Keep practicing these expansion techniques to reinforce your skills in logarithmic computations.

Power Rule of Logarithms

The power rule of logarithms is a fundamental property that helps simplify expressions where a logarithm is raised to a power. The power rule can be stated as: \( \ln (m^n) = n \ln m \). This rule allows you to bring the exponent down in front of the log, turning multiplication into a simpler expression involving addition or subtraction of logs.

In the example \( \ln x^{5/2} \), the power rule is applied to move the exponent \( 5/2 \) in front of the logarithm, making it \( \frac{5}{2} \ln x \). This application of the power rule takes log expressions with exponents and makes them linear in nature.

The power rule is not only useful for algebraic manipulations but also critical when dealing with calculus problems involving logs such as differentiation and integration. It's a versatile tool that enhances your ability to work with logarithmic equations more effectively.

Logarithmic Transformations

Logarithmic transformations are operations that employ the properties of logarithms to convert a given expression into a different format. These transformations are essential in simplifying calculations and solving equations.

By using transformations like rewriting a square root in terms of its corresponding exponent or utilizing rules such as the power, product, or quotient rules, we manage to streamline complex problems. These operations can help linearize exponential growth patterns, making them easier to interpret or solve.

Such transformations are often found in fields like statistics, where they're used to stabilize variance, and in computer science, for operations such as data compression. In this exercise, recognizing the square root as a power allowed us to employ the power rule, simplifying \( \ln \sqrt{x^5} \) into the more straightforward form \( \frac{5}{2} \ln x \). Grasping these transformations will strengthen your ability to handle logs in various mathematical contexts.