Problem 73 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. \(\cdot \frac{1}{4} \log _{3} 2+\frac{1}{4} \log _{3} x\)

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{1}{4} log_3 (2x)\)

Step 1: Recognize the common logarithm property

Start by identifying that both terms have a common base of 3, which means we can apply the property of logarithms \(log_b mn = log_b m + log_b n\)

Step 2: Apply the property of multiplication to the two logarithms

Applying the property of multiplication, the equation \(\frac{1}{4} log_3 2 + \frac{1}{4} log_3 x\) transforms into \(\frac{1}{4} log_3 (2x)\)

Step 3: Check and present the final answer

Check to make sure that the final expression is formatted correctly. The final result should be \(\frac{1}{4} log3 (2x)\)

Key Concepts

Logarithm Properties

Logarithm properties are essential tools in simplifying complex logarithmic expressions. They help us transform equations into more manageable forms. One of the most fundamental properties is the **Product Property**. This property states that the logarithm of a product is the sum of the logarithms of its factors. Mathematically, it is expressed as: \ \[\log_b(mn) = \log_b m + \log_b n\] This property is particularly useful when dealing with logarithms that have the same base, as it allows us to combine them into a single expression.

• **Product Property**: \(\log_b(mn) = \log_b m + \log_b n\)
• **Quotient Property**: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\)
• **Power Property**: \(\log_b(m^n) = n \cdot \log_b(m)\)

In the given exercise, we use the product property to combine two logarithms with a common base of 3 into a single expression. Recognizing these properties and how to apply them correctly is key to simplifying logarithmic expressions efficiently.

Single Logarithm

Writing an expression as a single logarithm is an effective technique to simplify equations and make them easier to handle. The objective is to consolidate multiple logarithm terms into one single logarithmic expression. This process often involves using logarithm properties, such as the product, quotient, and power rules. In our example, we start with multiple terms \(\frac{1}{4} \log_3 2 + \frac{1}{4} \log_3 x\). The first step is recognizing that both terms share a common factor of \(\frac{1}{4}\). Hence, we apply the product property to combine the logarithms: \ \[\frac{1}{4} \log_3 2 + \frac{1}{4} \log_3 x = \frac{1}{4} \log_3 (2x)\] Simplifying expressions into a single log form makes the manipulation and computation much more straightforward, especially when solving equations or evaluating functions. Remember, working toward a single logarithm always involves looking for common bases and combining terms appropriately.

Base of Logarithm

The base of a logarithm is an important feature that defines the logarithmic function. It indicates the number to which you raise to obtain another number. The base must always be a positive number and different from 1. Common logarithm bases include:

• **Base 10** (common logarithm, \(\log\))
• **Base \(e\)** (natural logarithm, \(\ln\))
• **Base 2** (binary logarithm, \(\log_2\))

In some exercises, the base plays a critical role, especially when applying properties like the change of base formula or when solving equations. In our task, the base is 3, a less common but crucial base for the problem at hand. When simplifying logarithms or applying logarithm properties, always verify that the base remains consistent across terms.Understanding the base helps ensure accuracy in transforming expressions and solving logarithmic equations. Different bases might require specific rules or conversions, making it essential to grasp the influence each base has on the logarithmic relationship.