Problem 71 - Intermediate Algebra

Question

For Problems \(69-80\), express each of the following as the sum or difference of simpler logarithmic quantities. Assume all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(\frac{y}{z}\right) $$

Step-by-Step Solution

Answer

The expression is \( \log_b y - \log_b z \).

Step 1: Apply the Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). Here, identify \( M = y \) and \( N = z \). So, we can write: \[ \log_b \left( \frac{y}{z} \right) = \log_b y - \log_b z \].

Step 2: Simplify the Expression

Since there are no additional simplifications for \( \log_b y \) and \( \log_b z \), the expression is as simplified as it can be following the application of the quotient rule. The expression \( \log_b \left( \frac{y}{z} \right) \) simplifies to \( \log_b y - \log_b z \).

Key Concepts

Quotient Rule of Logarithms

When working with logarithms, you may encounter the need to simplify expressions like divisions inside logarithmic terms. The quotient rule of logarithms provides a handy way to manage these by allowing us to break down complex divisions into simpler subtraction expressions between logs.

This rule states that for any positive real numbers \( M \) and \( N \), and a positive base \( b \), the logarithm of a quotient can be expressed as a difference:

• \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)

The idea here is that division within a logarithm can be turned into subtraction, making it easier to work with. This step helps in transforming an expression into something more familiar and manageable, especially if you have to plug in values or perform algebraic manipulations later on.

For instance, when given \( \log_b \left( \frac{y}{z} \right) \), you directly apply the rule to obtain \( \log_b y - \log_b z \). With this technique, you can handle more complex logarithmic expressions effectively.

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms, which are the inverse operations of exponents. They enable us to solve equations or simplify expressions where the variable is an exponent.

When dealing with logarithmic expressions, you often need to follow specific rules and laws of logarithms to rewrite them in a simpler or more useful form. Beyond the quotient rule, here are a few other common rules of logarithms that you might encounter:

• Product Rule: \( \log_b (MN) = \log_b M + \log_b N \)
• Power Rule: \( \log_b (M^k) = k \cdot \log_b M \)

These logarithmic properties are central in simplifying and solving problems involving logarithms. Understanding how to manipulate these expressions grants you the flexibility needed in various mathematical scenarios, from algebra to calculus.

Therefore, having a firm grasp on logarithmic expressions and their properties is crucial for further exploration of mathematics and its countless applications.

Simplifying Logarithms

Simplifying logarithms is the process of using known logarithmic properties to transform and reduce complex expressions to a more concise form. This often involves changing the expression into a sum, difference, or multiplied form, which often makes further calculations simpler and more intuitive.

Let's take the expression \( \log_b \left( \frac{y}{z} \right) \) as an example. Using the quotient rule of logarithms, it can be simplified to \( \log_b y - \log_b z \). This simplification has practical benefits:

• It eases the calculation, especially when evaluating the expression numerically.
• It provides clearer insight into how each component of the original expression contributes to the result.
• It aids in recognizing patterns or symmetries that are not immediately apparent.

Simplifying logarithms often includes evaluating whether any further reductions or transformations can be performed, such as using the product or power rules. However, it’s important to note that once you reach an expression that can't be reduced by known rules, you have simplified it as much as possible.

Always check each component within a logarithmic expression individually to ensure it cannot be further broken down. By mastering simplification, you open up pathways to solving more complex mathematical problems efficiently.