Problem 17 - Algebra 2 and Trigonometry

Question

a. Write each expression as a single logarithm. b. Find the value of each expression. \(\log _{3} 243-\log _{3} 729\)

Step-by-Step Solution

Answer

\( \log_{3} 243 - \log_{3} 729 = -1 \).

Step 1: Understanding the Expression

We have the expression \( \log_{3} 243 - \log_{3} 729 \). This is a difference of logarithms with the same base, allowing us to use properties of logarithms.

Step 2: Applying Logarithm Properties

Use the property \( \log_{a} b - \log_{a} c = \log_{a} \left( \frac{b}{c} \right) \). This means we can rewrite \( \log_{3} 243 - \log_{3} 729 \) as a single logarithm: \( \log_{3} \left( \frac{243}{729} \right) \).

Step 3: Simplifying the Fraction

Calculate \( \frac{243}{729} \). Both numbers divide exactly, as 243 is \( 3^5 \) and 729 is \( 3^6 \). Thus, \( \frac{3^5}{3^6} \) simplifies to \( 3^{-1} \), or \( \frac{1}{3} \).

Step 4: Express as Single Logarithm

The expression simplifies further to \( \log_{3} \left( \frac{1}{3} \right) \), which according to the properties of logarithms, simplifies to \( -1 \) because it represents the exponent needed to raise 3 to get \( \frac{1}{3} \).

Step 5: Calculating the Value

Thus, the value of the expression \( \log_{3} \left( \frac{1}{3} \right) \) is \(-1\).

Key Concepts

Properties of Logarithms

Logarithms have several useful properties that allow us to manipulate and simplify expressions. Understanding these properties makes it easier to solve problems involving logarithms. Let's break down some of the main properties:

• **Product Property**: \( \log_{a}(b \times c) = \log_{a}b + \log_{a}c \). This tells us that the logarithm of a product is the sum of the logarithms.
• **Quotient Property**: \( \log_{a} \left( \frac{b}{c} \right) = \log_{a}b - \log_{a}c \). We use this property when dealing with expressions where one logarithm is subtracted from another with the same base.
• **Power Property**: \( \log_{a}(b^c) = c \times \log_{a}b \). This property allows you to bring the exponent in front, turning multiplication into a simpler addition operation.

In our exercise, we used the Quotient Property to simplify \( \log_{3}243 - \log_{3}729 \) into \( \log_{3} \left( \frac{243}{729} \right) \). Recognizing when these properties apply is key to efficiently solving logarithmic expressions.

Simplification of Expressions

Simplification is about making expressions as basic as possible by using mathematical rules and properties. For logarithmic expressions, simplification frequently involves breaking down complex parts into singular forms. This process often uses prime factorization, as seen in the given problem.
In the exercise, we calculated \( \frac{243}{729} \) to simplify. Observing that 243 and 729 can be represented as powers of 3, specifically \(3^5\) and \(3^6\) respectively, allowed us to turn the fraction into \( \frac{3^5}{3^6} \). Then, we further simplified it by applying exponent subtraction, resulting in the simpler \(3^{-1} = \frac{1}{3} \).
Simplifying also helps us express a logarithm with a single term, making calculations direct and easier to interpret.

Logarithmic Expressions

A logarithmic expression is an important tool in mathematics that represents exponents. In dealing with logs, it's crucial to recognize how these expressions convert complex multiplicative tasks into manageable additive terms or transform divisions into subtractions.

• **Base**: The small number acting as the foundation for the expression, determining the factor of multiplication.
• **Argument**: The number we are interested in getting through repeated multiplication using the base.

In our exercise, we dealt with the expression \( \log_{3} \left( \frac{1}{3} \right) \). Understanding that the goal of a logarithmic expression is to find the exponent that the base needs to achieve the argument allows us to solve it effectively. We interpreted it as the value \(-1\) because \(3^{-1}\) equals \( \frac{1}{3} \). This clarity in interpreting the expression makes logarithmic calculations clearer and efficient, helping us in both academic and practical applications.