Problem 75 - Algebra for College Students

Question

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(\frac{x^{1 / 2} y^{1 / 3}}{z^{4}}\right) $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \log_{b} x + \frac{1}{3} \log_{b} y - 4 \log_{b} z \).

Step 1: Identify the Logarithmic Expression

We are given the expression \( \log_{b}\left(\frac{x^{1/2} y^{1/3}}{z^{4}}\right) \). Our task is to express this as a sum or difference of simpler logarithmic quantities.

Step 2: Apply the Quotient Rule of Logarithms

Start by applying the quotient rule of logarithms: \( \log_{b}\left(\frac{u}{v}\right) = \log_{b} u - \log_{b} v \). This transforms the expression into \( \log_{b}\left(x^{1/2} y^{1/3}\right) - \log_{b}\left(z^{4}\right) \).

Step 3: Apply the Product Rule of Logarithms

The term \( \log_{b}\left(x^{1/2} y^{1/3}\right) \) can be split using the product rule: \( \log_{b}(uv) = \log_{b} u + \log_{b} v \). Thus, \( \log_{b}\left(x^{1/2}\right) + \log_{b}\left(y^{1/3}\right) \).

Step 4: Use the Power Rule of Logarithms

Apply the power rule \( \log_{b}(x^{m}) = m \log_{b}(x) \) to each term: \( \frac{1}{2} \log_{b} x + \frac{1}{3} \log_{b} y \) and subtract \( 4 \log_{b} z \).

Step 5: Combine All the Steps

Combine all expressions: \( \frac{1}{2} \log_{b} x + \frac{1}{3} \log_{b} y - 4 \log_{b} z \). This is the expression as the sum and difference of simpler logarithmic quantities.

Key Concepts

Logarithm Properties

When working with logarithmic expressions, it's essential to understand the fundamental properties that govern them. Logarithms are the inverse operations to exponentiation, and they simplify the handling of powers and roots. Some primary properties include:

• The **Product Rule**, which allows us to split a logarithm of a product into a sum of logarithms.
• The **Quotient Rule**, which helps us divide the logarithm of a fraction into a difference.
• The **Power Rule**, which aids in bringing down exponents within logarithms as coefficients.

These properties make it easier to manipulate logs and transform them into more manageable expressions. Viewed in this way, logarithms are not only tools for solving equations but also ways to simplify complex algebraic expressions.

Quotient Rule

The **Quotient Rule** for logarithms is the idea that the logarithm of a division can be expressed as the subtraction of two logarithms. Formally, this is written as \( \log_{b}\left(\frac{u}{v}\right) = \log_{b} u - \log_{b} v \).
This property is useful when you need to break down a logarithmic expression that involves a fraction. The rule simplifies the expression by allowing us to split the log into two parts, which are usually easier to work with.
For instance, in the exercise where we have \( \log_{b}\left(\frac{x^{1/2} y^{1/3}}{z^{4}}\right) \), applying the Quotient Rule gives us: \( \log_{b}(x^{1/2} y^{1/3}) - \log_{b}(z^{4}) \). This simplifies the initial complex expression into more manageable parts.

Product Rule

The **Product Rule** is a property that helps to split the logarithm of a product into a sum of separate logarithms. The formal expression for the Product Rule is \( \log_{b}(uv) = \log_{b} u + \log_{b} v \).
This rule comes in handy when dealing with logarithms of products, making them simpler by creating a sum of individual logarithms.
In our exercise, after applying the Quotient Rule, we had \( \log_{b}(x^{1/2} y^{1/3}) \). By using the Product Rule, we further split this into \( \log_{b}(x^{1/2}) + \log_{b}(y^{1/3}) \). This transformation makes it easier to apply the Power Rule later on and to handle each logarithm separately.

Power Rule

The **Power Rule** is a logarithmic property that allows you to take exponents out of a logarithm and turn them into coefficients. It is expressed as \( \log_{b}(x^{m}) = m \log_{b}(x) \).
This rule is particularly powerful when dealing with expressions that involve powers, as it simplifies them significantly.
In the exercise, after applying both the Quotient and Product Rules, we have expressions like \( \log_{b}(x^{1/2}) \) and \( \log_{b}(y^{1/3}) \). Applying the Power Rule turns these into \( \frac{1}{2} \log_{b} x \) and \( \frac{1}{3} \log_{b} y \).
Finally, we also apply the Power Rule to \( \log_{b}(z^{4}) \), obtaining \( 4 \log_{b} z \). This results in the entire expression: \( \frac{1}{2} \log_{b} x + \frac{1}{3} \log_{b} y - 4 \log_{b} z \), beautifully simplified and separated.