Problem 47 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions. $$\frac{1}{2} \ln x+\ln y$$

Step-by-Step Solution

Answer

\(\frac{1}{2} \ln x + \ln y\) can be simplified to \(\ln(\sqrt{x}y)\).

Step 1: Adjust the Coefficient

Initially, observe that the first term has a coefficient of \(1/2\), so the goal is to adjust this term to have a coefficient of '1'. Use the power rule of logarithms, which states that \(\log_b M^p = p \log_b M\), to rewrite the first term: \(\frac{1}{2} \ln x = \ln (\sqrt{x})\).

Step 2: Combine Log Expressions

Now that both terms have a coefficient of 1, apply the product rule of logorithms, \(\log_b M + \log_b N = \log_b (M*N)\), to combine the two terms into a single logarithmic expression. This results in: \(\ln(\sqrt{x}) + \ln (y) = \ln(\sqrt{x}y)\).

Step 3: Simplify the Log Expression

Check whether the expression \(\sqrt{x}y\) can be further simplified. Since there is no particular restriction mentioned about \(x\) and \(y\), the current form of the single logarithm is the simplest form.

Key Concepts

Logging Rules

Understanding the rules of logarithms, often referred to as 'logging rules', is crucial for simplifying complex logarithmic expressions. A logarithm, at its most fundamental level, is really another way to express exponentiation. It tells us what power a certain number (the base) must be raised to give another number. For instance, the expression \( \log_b x = y \) implies that \( b^y = x \.\)

There are several logging rules that aid in the manipulation of logs:

• The Product Rule: \( \log_b(MN) = \log_b M + \log_b N \), which allows us to split the log of a product into a sum of two logs.
• The Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \), which lets us restate the log of a quotient as a difference of logs.
• The Power Rule: \( \log_b M^p = p \log_b M \), a powerful tool for moving exponents in and out of the log function.

These rules lay the foundation for converting complex logarithmic expressions into simpler forms and vice versa, facilitating deeper understanding and easier computation.

Condensing Logarithms

Condensing logarithms means to combine multiple logarithmic terms into a single term with a coefficient of 1, often with the goal of simplification or further manipulation. To condense logs, you usually move in reverse from the 'logging rules' by applying them from right to left. Let's take the given exercise \( \frac{1}{2} \ln x +\ln y \) as an example:

Following the previously mentioned power rule, first adjust coefficients that are not 1. Convert \(\frac{1}{2} \ln x\) into \(\ln(\sqrt{x})\), as shown in the textbook solution. After achieving coefficients of 1, employ the product rule in reverse. Instead of splitting a log of a product into a sum, you combine the sum of logs into a single log that expresses a product—\(\ln(\sqrt{x}y)\). This serves to condense the original expression effectively and often paves the way towards further evaluation or application in various mathematical problems.

Logarithmic Expressions

A logarithmic expression involves logs, including numbers, variables, and operations within the log function. These expressions can range from the very simple, like \( \log_b x \), to the very complex, involving multiple terms and operations. As seen in the previous sections, the various logging rules are necessary 'tools' to handle these expressions.

Handling such expressions skillfully often requires a series of steps: identifying the terms, checking if the rules of logarithms apply, performing the appropriate restructuring using these rules—whether expanding or condensing—and then simplifying the resulting expression. It's also important to be aware of the domain of the variables within logarithmic expressions, as logs of negative numbers or zero are undefined in real numbers. Whether you're working through homework or encountering logarithms in higher-level mathematics, being familiar with these expressions and their manipulations is essential for your toolkit in algebra and calculus.