Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log \left(x^{2} y^{3} \sqrt[3]{x^{2} y^{5}}\right)\)

Step-by-Step Solution

Answer

\( \frac{8}{3} \log(x) + \frac{14}{3} \log(y) \)

Step 1: Identify the Expression Inside the Logarithm

The expression inside the logarithm is \( x^{2} y^{3} \sqrt[3]{x^{2} y^{5}} \), which means we need to expand \( \log(x^{2} y^{3} \sqrt[3]{x^{2} y^{5}}) \). This expression involves products, powers, and a root. We'll apply logarithm properties to expand it.

Step 2: Apply the Product Property

The product property of logarithms states \( \log(AB) = \log A + \log B \). First, split the complex expression into a sum of logs: \( \log(x^{2} y^{3} \sqrt[3]{x^{2} y^{5}}) = \log(x^{2} y^{3}) + \log(\sqrt[3]{x^{2} y^{5}}) \).

Step 3: Apply the Product Property Again

Further break down \( \log(x^{2} y^{3}) \) using \( \log(AB) = \log A + \log B \). This becomes \( \log(x^{2}) + \log(y^{3}) \).

Step 4: Use Power Property on Separate Logs

Now apply the power property \( \log(A^{c}) = c \log A \) to \( \log(x^{2}) \) and \( \log(y^{3}) \):- \( \log(x^{2}) = 2 \log(x) \)- \( \log(y^{3}) = 3 \log(y) \).

Step 5: Expand the Root Expression

The expression \( \log(\sqrt[3]{x^{2} y^{5}}) \) contains a cube root, which can be rewritten using exponents: \((x^{2} y^{5})^{1/3}\). Applying the power property, we get \( \frac{1}{3} \log(x^{2} y^{5}) \).

Step 6: Apply the Product Property to Root Expression

Expand \( \log(x^{2} y^{5}) \) inside the root expression using the product property: \( \log(x^{2} y^{5}) = \log(x^{2}) + \log(y^{5}) \).

Step 7: Use Power Property Again

Apply the power property inside the square root term:- \( \log(x^{2}) = 2 \log(x) \)- \( \log(y^{5}) = 5 \log(y) \). Depending on the expression \( \log(x^{2} y^{5}) = 2 \log(x) + 5 \log(y) \).

Step 8: Combine the Expanded Logs

Now, multiply each term by \( \frac{1}{3} \) due to the cube root: \(\frac{1}{3}(2 \log(x) + 5 \log(y)) = \frac{2}{3} \log(x) + \frac{5}{3} \log(y) \).Finally, combine all expanded terms from previous steps: \( 2 \log(x) + 3 \log(y) + \frac{2}{3} \log(x) + \frac{5}{3} \log(y) \).

Step 9: Simplify the Expression

Group similar logarithmic terms together to get:\[ 2 \log(x) + \frac{2}{3} \log(x) + 3 \log(y) + \frac{5}{3} \log(y) \]Combine them to yield:\[ \left( 2 + \frac{2}{3} \right) \log(x) + \left( 3 + \frac{5}{3} \right) \log(y) \]Simplify further by converting the fractions:\[ \frac{6}{3} + \frac{2}{3} = \frac{8}{3} \text{ and } \frac{9}{3} + \frac{5}{3} = \frac{14}{3} \]So, the expression becomes \[ \frac{8}{3} \log(x) + \frac{14}{3} \log(y) \].

Key Concepts

Logarithmic Expansion

Logarithmic expansion is a useful technique when dealing with complex logarithmic expressions. The aim is to rewrite a logarithm into a more manageable form, often expressed as a sum, difference, or product of simpler logs. This is achieved by applying the properties of logarithms, such as the product and power properties. By expanding a logarithm, you're breaking down a complicated problem into smaller, more comprehensible pieces.

In the exercise provided, we dealt with the expansion of \( \log (x^{2} y^{3} \sqrt[3]{x^{2} y^{5}} ) \). This expression involves both powers and roots as well as products. Each of these elements can be individually addressed using the specific properties of logs to simplify the overall expression. Through expansion, grasping the underlying components becomes not only easier but also reveals insights into solving similar problems effectively.

Product Property

The product property of logarithms is a cornerstone concept. It allows one to break down a logarithm of a product into a sum of logs. Mathematically, it is expressed as:\(\log(AB) = \log A + \log B\). This property is extensively used in simplifying logarithmic expressions.

In the exercise, this property was crucial. The task was to expand \( \log (x^{2} y^{3} \sqrt[3]{x^{2} y^{5}}) \). Initially, it was split using the product property into:

• \( \log (x^{2} y^{3}) \)
• \( \log (\sqrt[3]{x^{2} y^{5}}) \)

Subsequent application of this property further simplified \( \log (x^{2} y^{3})\) into \( \log(x^{2}) + \log(y^{3}) \). By breaking down the complex logarithmic expression into a series of simpler parts, the solution process becomes much more straightforward and organized.

Power Property

The power property of logarithms plays a key role when one needs to remove exponents from expressions within a logarithm. It translates an exponent into a coefficient, allowing easier manipulation. Expressed mathematically, the power property states:\(\log(A^{c}) = c \log A\).

Applying this property was crucial in our task to expand \( \log (x^{2} y^{3} \sqrt[3]{x^{2} y^{5}})\). Each part of the expression containing powers was simplified:

• \( \log(x^{2}) = 2 \log(x) \)
• \( \log(y^{3}) = 3 \log(y) \)
• \( \frac{1}{3} \log(x^{2}) = \frac{2}{3} \log(x) \)
• \( \frac{1}{3} \log(y^{5}) = \frac{5}{3} \log(y) \)

This transformation is essential in changing exponential components into linear terms, helping to simplify and expand complex logarithmic expressions.

Simplification of Logarithmic Expressions

Simplification is the final and often the most satisfying step in logarithmic problems. It involves combining like terms and reducing the expression to its simplest form. Here, careful application of both the product and power properties comes to fruition.

In the provided problem, after expanding \( \log (x^{2} y^{3} \sqrt[3]{x^{2} y^{5}})\) into its components, terms were combined to yield:\[\left( 2 + \frac{2}{3} \right) \log(x) + \left( 3 + \frac{5}{3} \right) \log(y) \].
By simplifying the coefficients, the final form was:\[\frac{8}{3} \log(x) + \frac{14}{3} \log(y) \].
Simplification makes the expression not only cleaner but also easy to interpret. This final transformation often provides deeper insight into the mathematical context of a problem.