Problem 85 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt[4]{x^{3}\left(x^{2}+3\right)}\)

Step-by-Step Solution

Answer

The expanded form of the given logarithm expression is \(\frac{3}{4} \ln x + \frac{1}{2} \ln x + \frac{1}{4}\ln 3\)

Step 1: Apply the root rule

The logarithm of a root can be simplified by moving the root as a fraction in front of the logarithm. \(\ln \sqrt[4]{x^{3}\left(x^{2}+3\right)}\) will become \(\frac{1}{4} \ln \left(x^{3}(x^{2}+3) \right)\)

Step 2: Apply the product rule

The product rule allows us to split a logarithm that involves multiplication into the sum of two logs. So, we can split \(\frac{1}{4} \ln \left(x^{3}(x^{2}+3) \right)\) into \(\frac{1}{4}( \ln x^{3} + \ln \left(x^{2}+3\right))\)

Step 3: Apply the power rule

According to the power rule, logarithm of a power can be rewritten by moving the exponent in front of the log. So \(\frac{1}{4}( \ln x^{3} + \ln \left(x^{2}+3\right))\) turns into \(\frac{3}{4} \ln x + \frac{1}{4} \ln \left(x^{2}+3\right)\)

Step 4: Apply the power rule again

We can use the power rule again on the term \(\ln \left(x^{2}+3\right)\) and the whole expression simplifies to \(\frac{3}{4} \ln x + \frac{1}{2} \ln x + \frac{1}{4}\ln 3\)

Key Concepts

Natural Logarithm Expansion

Expanding logarithms, particularly the natural logarithm (denoted as \( \ln \)), is an essential skill in mathematics that allows students to simplify complex expressions for easier computation or to find derivatives and integrals in calculus. Consider the expression \( \ln \sqrt[4]{x^{3}(x^{2}+3)} \). To expand this expression, we need to apply logarithmic properties step by step. First, we address the fourth root by invoking the logarithmic root rule, transforming the root into a fraction in front of the logarithm and yielding \( \frac{1}{4} \ln (x^{3}(x^{2}+3)) \).

Understanding this concept is crucial for mathematical manipulations involving logarithms, as it enables you to rewrite expressions in a form where the properties of logarithms can be readily applied.

Logarithmic Product Rule

The logarithmic product rule is a powerful tool that comes into play when you are dealing with the logarithm of a product, such as \( x^{3}(x^{2}+3) \). According to this rule, the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Breaking Down Products

• Start with the product inside a logarithm.
• Separate the product into individual logarithms.
• Add these logarithms together.

Applying the product rule to our expression, we split the logarithm as \( \frac{1}{4} (\ln x^{3} + \ln (x^{2}+3)) \). This simplification turns the tough task of finding the logarithm of a complex product into something much more manageable.

Logarithmic Power Rule

The logarithmic power rule simplifies the process of working with logarithms of powers. In essence, it allows us to take the exponent of the argument and multiply it directly by the logarithm, converting \( \ln x^{n} \) to \( n\ln x \).

Take the term \( \ln x^{3} \) from our previous step. By applying the power rule, it becomes \( 3\ln x \), effectively pulling out the power and placing it in front of the logarithm. Through the use of this rule, we greatly simplify expressions involving exponents within logarithms, such as turning our entire expression into \( \frac{3}{4} \ln x + \frac{1}{4} \ln (x^{2}+3) \).

Logarithmic Root Rule

The logarithmic root rule is closely related to the power rule and is particularly useful when dealing with the logarithm of a root expression. Much like the product rule, this rule helps in breaking down more complex logarithms into simpler parts. It states that the logarithm of a root is equal to the logarithm of the radicand divided by the index of the root: \( \ln \sqrt[n]{x} = \frac{1}{n} \ln x \).

In our exercise, we applied this rule to the fourth root \( \sqrt[4]{x^{3}(x^{2}+3)} \), which simplified the expression significantly. Knowing when and how to apply the logarithmic root rule is key to managing expressions with roots in the realm of logarithms.