Problem 84 - Algebra for College Students

Question

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ \left(\log _{b} x-\log _{b} y\right)-\log _{b} z $$

Step-by-Step Solution

Answer

The single logarithm is \(\log_b (\frac{x}{yz})\)."

Step 1: Identify the Given Expression

We need to express \((\log_b x - \log_b y) - \log_b z\) as a single logarithm. This is a subtraction among three logarithmic terms.

Step 2: Apply the Logarithmic Subtraction Rule

The logarithmic subtraction rule states that \(\log_{b}(a) - \log_{b}(c) = \log_{b}(\frac{a}{c})\). Apply this rule first to the inner expression: \(\log_{b} x - \log_{b} y\) which becomes \(\log_{b}(\frac{x}{y})\).

Step 3: Apply the Rule Again to the Result

Now, apply the logarithmic subtraction rule to \((\log_{b}(\frac{x}{y})) - \log_{b} z\). This becomes \(\log_{b}(\frac{\frac{x}{y}}{z})\).

Step 4: Simplify the Nested Fraction

Simplify the nested fraction to a single expression: \(\frac{x/y}{z} = \frac{x}{yz}\). Therefore, the expression becomes \(\log_b (\frac{x}{yz})\).

Key Concepts

Properties of Logarithms

Logarithms are incredibly useful, and they come with a set of properties that make dealing with them much easier. These properties, often called the laws of logarithms, allow us to manipulate logarithmic expressions to make them simpler or to combine them into single logarithmic terms. The most common properties include the product rule, quotient rule, and power rule.

• Product Rule: \(\log_b (mn) = \log_b m + \log_b n\). This allows us to turn the logarithm of a product into the sum of two separate logarithms.
• Quotient Rule: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\). With this rule, you can express the logarithm of a quotient as the difference between two logarithms.
• Power Rule: \(\log_b (m^k) = k \cdot \log_b m\). This simplifies the logarithm of a power into the product of the exponent and the logarithm of the base.

By using these properties, we can often reduce complex logarithmic expressions into much simpler forms, making our calculations more efficient. For example, when we want to express a combination of logs as a single log, these properties are the tools to do so.

Logarithmic Subtraction

Logarithmic subtraction is a direct application of one of the core properties of logarithms: the quotient rule. As previously mentioned, this rule allows us to express the subtraction of two logs as a single log of a quotient.When you encounter an expression like \(\log_b x - \log_b y\), it can be rewritten using the quotient rule as \(\log_b \left(\frac{x}{y}\right)\). In essence, you're not only simplifying the expression but also condensing two logs into one, which can be particularly useful in solving logarithmic equations or inequalities.

• This method helps to identify common factors in numerators and denominators, paving the way for further simplification.
• Logarithmic subtraction emphasizes a critical aspect of logarithms in relation to division, which is central to many logarithmic manipulations.

In practical problems, such as the original exercise given, this subtraction rule is applied iteratively to reduce a complex expression into a more manageable form.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using known rules and properties to make a complex expression more manageable. In many textbook exercises, the goal is to express a given formula as a single logarithm -- making the expression cleaner and often revealing deeper insights into the relationships between the variables involved.In our sample exercise, we started with the expression \((\log_b x - \log_b y) - \log_b z\). The simplification process is:

• First, apply the quotient rule to the inner expression: \(\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right)\).
• Next, bring in the third term using the quotient rule again: \((\log_b \left(\frac{x}{y}\right) - \log_b z) = \log_b \left(\frac{x}{yz}\right)\).
• The final expression \(\log_b \left(\frac{x}{yz}\right)\) represents the simplified form, achieved by turning subtractions into divisions and thus achieving the single-log expression goal.

Each step's goal is to apply logarithmic laws in reducing terms, simplifying the calculation process, or preparing for further steps in solving broader mathematical problems.