Problem 72 - Algebra A Combined Function

Question

Solve. Which of the following is the correct way to rewrite \(\log _{9} \frac{21}{3} ?\) a. \(\log _{9} 7\) b. \(\log _{9}(21-3)\) c. \(\frac{\log _{9} 21}{\log _{9} 3}\) d. \(\log _{9} 21-\log _{9} 3\)

Step-by-Step Solution

Answer

The correct answer is option d: \(\log_9 21 - \log_9 3\).

Step 1: Understand the Properties of Logarithms

We start with the given expression \(\log_9 \frac{21}{3}\). The properties of logarithms tell us that \(\log_b \frac{m}{n} = \log_b m - \log_b n\). This property will help us simplify the given logarithmic expression.

Step 2: Apply the Quotient Rule of Logarithms

Using the quotient rule \(\log_b \frac{m}{n} = \log_b m - \log_b n\), we can rewrite \(\log_9 \frac{21}{3}\) as \(\log_9 21 - \log_9 3\). This lets us express the logarithm of a fraction as a difference of two logarithms.

Step 3: Find the Correct Option

Now we compare our result \(\log_9 21 - \log_9 3\) with the options provided: a) \(\log_9 7\), b) \(\log_9 (21-3)\), c) \(\frac{\log_9 21}{\log_9 3}\), and d) \(\log_9 21 - \log_9 3\). The expression from Step 2 aligns exactly with option d.

Key Concepts

Properties of Logarithms

Logarithms are a fascinating mathematical concept, and understanding their properties is key to mastering them. One of the fundamental properties of logarithms is their ability to transform expressions through simplification rules, akin to the rules of exponents. The properties help us break down complex expressions into simpler ones.

A basic property to remember is the product property, \[ \log_b(m imes n) = \log_b m + \log_b n \]which explains that the logarithm of a product is the sum of the logarithms of the factors. Similarly, the quotient property is crucial and states:

• \( \log_b \frac{m}{n} = \log_b m - \log_b n \)

This property is particularly useful when simplifying the logarithm of a fraction by splitting it into a difference. These properties enable a deeper and more efficient approach to solving logarithmic expressions. Grasping these concepts can vastly simplify your math problems.

Quotient Rule of Logarithms

The quotient rule of logarithms is essential when dealing with division inside a logarithmic expression. It makes use of the property we introduced earlier:

• \( \log_b \frac{m}{n} = \log_b m - \log_b n \)

This rule states that the logarithm of a fraction is equal to the difference between the logarithm of the numerator and the logarithm of the denominator.

Let's consider an example with real numbers. If you have an expression like \( \log_9 \frac{21}{3} \), applying the quotient rule allows you to rewrite it as \( \log_9 21 - \log_9 3 \). This transformation can make it much easier to solve logarithmic equations by breaking them into more manageable parts.

Understanding the quotient rule is particularly useful for visualizing the symmetry and balance within logarithmic operations. It not only simplifies computations but also aids in building a solid foundation for further logarithmic exploration.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions requires a good grasp of the properties of logarithms. The aim is to transform a complex logarithmic expression into a simpler, equivalent form, which often makes solving equations much easier. By using properties like the product and quotient rules, this transformation is achievable.

For instance, consider the expression \( \log_9 \frac{21}{3} \). By applying the quotient rule, we simplify it to \( \log_9 21 - \log_9 3 \). Such simplification is not just about rewriting expressions; it enables easier comparison and computation.

Simplifying logarithmic expressions is a skill that every math student should refine. It's not only crucial for tackling homework problems but also for a deeper understanding of how logarithms work. When simplified, problems become more approachable, and solutions are more easily found.