Problem 17 - 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. \(\log \left(\sqrt{x^{3} y^{-4}}\right)\)

Step-by-Step Solution

Answer

\( \frac{3}{2} \cdot \log x - 2 \cdot \log y \)

Step 1: Apply the Square Root Property

To begin with, recognize that the square root can be expressed as a fractional exponent. This allows us to rewrite \( \sqrt{x^3 y^{-4}} \) as \( (x^3 y^{-4})^{1/2} \). Then, apply the power of a power property: \( \log((x^3 y^{-4})^\frac{1}{2}) = \frac{1}{2} \cdot \log(x^3 y^{-4}) \).

Step 2: Simplify Using the Product and Quotient Properties

Expand \( \log(x^3 y^{-4}) \) using the product property of logarithms, which states \( \log(ab) = \log(a) + \log(b) \). Write: \( \log(x^3) + \log(y^{-4}) \). Then apply the quotient property on the second term: \( \log(x^3) - \log(y^4) \).

Step 3: Use the Power Property

Apply the power property of logarithms, which states \( \log(a^b) = b \cdot \log(a) \), to each term. For the first term, \( \log(x^3) = 3 \cdot \log x \). For the second term, \(-\log(y^4) = -4 \cdot \log y \).

Step 4: Combine the Expanded Terms

After applying the power property to each term, substitute back into the original expression: \( \frac{1}{2} \left( 3 \cdot \log x - 4 \cdot \log y \right) \). Distribute the \( \frac{1}{2} \) to both terms to get \( \frac{3}{2} \cdot \log x - 2 \cdot \log y \).

Key Concepts

Logarithmic Expansion

Logarithmic expansion makes it easier to work with complex logarithmic expressions. By expanding a logarithm, we turn it into a combination of simple logs instead of a single complex expression. For example, when we expand \( \log \left( \sqrt{x^{3} y^{-4}} \right) \), we transform it step by step into a linear form. This process relies on various properties of logarithms like the product, quotient, and power properties, enabling us to simplify and rearrange the expression. Once expanded, it's more understandable and easier to work with, particularly for solving and analyzing mathematical problems.

Product Property of Logarithms

The product property of logarithms is a handy tool for breaking down logarithms with multiplied variables. It states:

• \( \log(ab) = \log(a) + \log(b) \)

This means a single logarithm with a multiplied argument can be split into the sum of two separate logarithms. In our example, we initially deal with \( \log(x^3 y^{-4}) \), a single log of a product. By applying the product property, we separate it into \( \log(x^3) + \log(y^{-4}) \). Breaking down logarithmic terms using this property clarifies their structure and facilitates further simplification using other properties.

Quotient Property of Logarithms

The quotient property helps to handle divisions inside logarithms. It expresses a quotient of logs in terms of a difference. The formula is as follows:

• \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \)

This property is especially useful when dealing with negative exponents, like in our example. When we expanded \( \log(y^{-4}) \), using the fact that this can be rewritten as \( \log\left(\frac{1}{y^4}\right) \), the quotient property lets us split it into a subtraction: \(-\log(y^4)\). Thus, negative exponents are easily managed and converted using the quotient property.

Power Property of Logarithms

The power property of logarithms is essential for moving exponents outside of logarithms. It states:

• \( \log(a^b) = b \cdot \log(a) \)

This property allows the exponent in a logarithmic expression to become a multiple in front. In our work, we take \( \log(x^3) \) and rewrite it as \( 3 \cdot \log(x) \), and similarly \( -\log(y^4) \) becomes \( -4 \cdot \log(y) \). Applying this property makes expressions linear by transferring power terms into coefficients, which simplifies calculations and makes further steps more manageable.