Problem 103 - Elementary and Intermediate Algebra

Question

Which pair of expressions in each list are equivalent? a. \(\log (9 \cdot 3)\) b. \(\log 9 \cdot \log 3 \) c. \(\log 9+\log 3\)

Step-by-Step Solution

Answer

Expressions a and c are equivalent.

Step 1: Understand the Problem

We need to determine which pair of expressions are equivalent among options a, b, and c. The expressions involve logarithms and their properties.

Step 2: Apply Logarithmic Properties

Recall the logarithmic property: \( \log(ab) = \log a + \log b \). This rule will help us analyze and compare the given expressions.

Step 3: Analyze Expression a

Expression a is \( \log(9 \cdot 3) \), which can be simplified using the property from Step 2. Thus, \( \log(9 \cdot 3) = \log 9 + \log 3 \).

Step 4: Analyze Expression b

Expression b is \( \log 9 \cdot \log 3 \). This is simply the multiplication of two logarithms and does not directly match any logarithmic identity. Thus, it cannot be simplified using basic logarithmic rules.

Step 5: Analyze Expression c

Expression c is \( \log 9 + \log 3 \). This directly uses the sum rule of logarithms, which is equivalent to \( \log(9 \cdot 3) \).

Step 6: Compare the Expressions

From Steps 3, 4, and 5, expressions a and c are equivalent: \( \log(9 \cdot 3) = \log 9 + \log 3 \). Expression b is not equivalent to either a or c.

Key Concepts

Equivalent Expressions

Logarithmic expressions often look different at first glance, but they can sometimes represent the same value. These are known as equivalent expressions. For example, consider the question of whether two expressions like \( \log (9 \cdot 3) \) and \( \log 9 + \log 3 \) are equivalent. You can determine if they are by using the properties of logarithms, which allow you to transform one expression into another. If you can manipulate or adjust them to look the same or produce the same numerical result, then they are equivalent. Recognizing and creating equivalent expressions is fundamental in algebra and calculus because it helps solve complex problems by simplifying initial conditions or constraints.

Properties of Logarithms

Logarithmic properties are rules that simplify expressions and calculations involving logs. These properties are key to understanding how different logarithmic expressions relate to one another.

• **Product Property:** This states that \( \log(ab) = \log a + \log b \). It allows you to break down the log of a product into the sum of two logs.
• **Quotient Property:** This tells us that \( \log(\frac{a}{b}) = \log a - \log b \). It breaks down the log of a division into the difference of logs.
• **Power Property:** According to this rule, \( \log(a^b) = b \cdot \log a \). It helps convert the log of a power into a product of the exponent and the log.

By applying these properties, you can identify whether expressions like \( \log(9 \cdot 3) \) and \( \log 9 + \log 3 \) are equivalent or not. Understanding these properties helps not only in simplifying expressions but also in solving equations and inequalities that involve logarithms.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often involves applying logarithmic identities or properties to make the expression easier to handle. For example, when simplifying \( \log(9 \cdot 3) \), you apply the product property to separate it into two distinct terms: \( \log 9 + \log 3 \). This technique is useful in calculations and helps find equivalent expressions much more straightforwardly.
Here are some steps to follow when simplifying:

• Identify which property of logarithms can be applied to the expression.
• Rewrite the expression using the identified property. For instance, turn a product within the log into a sum of logs.
• Simplify further if possible, often by calculating known values, like numerical logs or further application of logarithmic properties.

Through these processes, you can simplify what once seemed like a complex expression into something far more manageable.