Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, condense to a single logarithm if possible. \(\log _{b}(28)-\log _{b}(7)\)

Step-by-Step Solution

Answer

\( \log_{b}(4) \)

Step 1: Identify the Logarithmic Properties

Recall the logarithmic identity: the difference of two logs with the same base can be written as the log of the quotient. Specifically, \( \log_{b}(A) - \log_{b}(B) = \log_{b}\left(\frac{A}{B}\right)\).

Step 2: Apply the Property to the Expression

Apply the quotient property to the given expression \( \log_{b}(28) - \log_{b}(7) \) using the identity identified in Step 1. This becomes \( \log_{b}\left(\frac{28}{7}\right) \).

Step 3: Simplify the Expression Inside the Logarithm

Calculate the division \( \frac{28}{7} \). The result is 4, so the expression simplifies to \( \log_{b}(4) \).

Step 4: Write the Final Condensed Expression

The given logarithmic expression \( \log_{b}(28) - \log_{b}(7) \) condenses to \( \log_{b}(4) \).

Key Concepts

Difference of Logarithms

In mathematics, understanding the difference of logarithms is crucial for simplifying complex expressions. When you subtract one logarithm from another, you are essentially comparing the orders of magnitude for two quantities. This operation follows a special rule called the "quotient property." For logarithms with the same base, the difference can be transformed into a single log of a quotient.

• Imagine you have two logarithms, both with the same base: \( \log_{b}(A) \) and \( \log_{b}(B) \).
• The expression \( \log_{b}(A) - \log_{b}(B) \) represents their difference.
• According to the laws of logarithms, this difference is equivalent to \( \log_{b}\left(\frac{A}{B}\right) \).

This outcome shows us that logarithmic operations align closely with basic arithmetical operations, demonstrating the strength and elegance of mathematical principles.

Quotient Property

The quotient property of logarithms is a handy tool that simplifies expressions involving division within logarithms. By understanding this property, you can transform multiple logarithmic terms into a more manageable form. Here's how it works:

• Let's consider the expression \( \log_{b}(A) - \log_{b}(B) \).
• Using the quotient property, the expression converts to \( \log_{b}\left(\frac{A}{B}\right) \).
• This property is particularly useful because it combines two logarithmic terms into one, making complex expressions less intimidating.

Remember, this property only applies when you have subtraction between logs that share the same base. It simplifies the arithmetic that you need to do inside the logarithm and offers a neat pathway to solve or condense logarithmic expressions.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is an essential part of working efficiently with logarithms. It involves breaking down complex or multiple log terms into a single expression that's easier to work with. Here’s a simple guide to simplifying:

• Begin by identifying any operations between the logs. For example, in our exercise, this is the subtraction \( \log_{b}(28) - \log_{b}(7) \).
• Apply relevant logarithmic properties, like the quotient property, to rewrite the expression as \( \log_{b}\left(\frac{28}{7}\right) \).
• Next, simplify the arithmetic inside the log expression. Calculate \( \frac{28}{7} \) to get 4.
• Finally, express your simplified logarithmic expression as \( \log_{b}(4) \).

By systematically using logarithmic rules and basic arithmetic, you can turn complex logarithmic expressions into simpler, more concise forms. This approach not only gives you cleaner results but also enhances your problem-solving skills in mathematics.