Problem 81 - 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.)\(\ln \frac{z}{\sqrt[3]{z+3}}\)

Step-by-Step Solution

Answer

The expanded form of the given logarithm is \(\ln z - \tfrac{1}{3} \ln (z + 3)\).

Step 1: Apply the Quotient Rule

The quotient rule for logarithms states that \(\ln(a/b) = \ln(a) - \ln(b)\). Apply this rule to separate the given expression into a difference of two logarithms: \(\ln \frac{z}{\sqrt[3]{z+3}} = \ln z - \ln \sqrt[3]{z + 3}\).

Step 2: Apply the Power Rule

The power rule denotes \(\ln a^k = k \ln a\) where \(k\) is any real number. Apply this rule to handle the cube root, recalling that \(\sqrt[3]{a} = a^{1/3}\): \(\ln \sqrt[3]{z + 3} = \tfrac{1}{3} \ln (z + 3)\). Thus, the expression becomes: \(\ln z - \tfrac{1}{3} \ln (z + 3)\).

Step 3: Rewrite the Final Expression

After applying both properties, gather your results into a single expression. The final simplified form is: \(\ln z - \tfrac{1}{3} \ln (z + 3)\).

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental logarithmic property that helps simplify expressions where a division inside a logarithm is involved. When you have a logarithm of the form \( \ln \left( \frac{a}{b} \right) \), you can break it down into a subtraction of two separate logarithms: \( \ln(a) - \ln(b) \). This means that instead of trying to handle a complex fraction within a logarithm, you recognize it as two simpler logarithmic expressions.

Here's why this is helpful:

• It transforms division into subtraction, simplifying calculations.
• Develops a deeper understanding of how multiplication and division relate to addition and subtraction in the logarithmic context.
• Reduces complex problems into more manageable parts.

For instance, in the expression \( \ln \left( \frac{z}{\sqrt[3]{z+3}} \right) \), applying the Quotient Rule helped divide it into two more straightforward terms: \( \ln z \) and \( \ln \sqrt[3]{z+3} \). This simplification step is crucial for further expanding or manipulating the expression.

Power Rule

The Power Rule is another important property of logarithms which helps you handle expressions with exponents more efficiently. It states that for any logarithm of the form \( \ln(a^k) \), you can bring the exponent down in front of the logarithm, obtaining \( k \ln a \). This property is incredibly useful when you need to simplify expressions involving powers.

In practice, this means:

• Exponentiation inside a logarithm can be counteracted by multiplication outside it.
• Makes it easier to expand and simplify complex expressions with exponents.

Consider the term \( \ln \sqrt[3]{z+3} \) from our example. Recognizing that a cube root is the same as an exponent of \( \frac{1}{3} \), we apply the Power Rule to simplify it to \( \frac{1}{3} \ln(z+3) \). Essentially, you leverage this rule to transform the expression into a more manageable form, making further manipulations more intuitive.

Logarithm Expansion

Logarithm Expansion involves using properties of logarithms, like the Quotient and Power Rules, to expand complex logarithmic expressions into simpler parts that add or subtract. This technique breaks down an expression so you can understand and calculate it more easily. The expansion is particularly advantageous when dealing with functions that have quotients, products, or powers, enabling you to express them as sums and differences in a clearer format.

The primary benefits of logarithm expansion include:

• Clarifies and simplifies complex expressions.
• Enables the breakdown of functions into simpler, more digestible components.
• Facilitates solving of logarithmic equations and inequalities.

Expanding \( \ln \frac{z}{\sqrt[3]{z+3}} \), as demonstrated with the given solution, illustrates how the original complex expression can be broken into \( \ln z - \frac{1}{3} \ln(z+3) \). Each part of the expansion becomes straightforward and easy to handle, thanks to the systematic application of logarithmic properties.