Problem 1 - Algebra and Trigonometry

Question

The logarithm of a product of two numbers is the same as the _____ of the logarithms of these numbers. So \(\log _{5}(25 \cdot 125)=\) _____ + _____ .

Step-by-Step Solution

Answer

sum, \( \log_5 25 \), \( \log_5 125 \).

Step 1: Understanding Logarithm Properties

In this exercise, we're dealing with a property of logarithms, specifically the property that expresses how the logarithm of a product can be split into a sum. The property is: if \( a \) and \( b \) are any two numbers, then \( \log_b(a \cdot c) = \log_b a + \log_b c \). This means the logarithm of a product is equal to the sum of the logarithms of the individual numbers.

Step 2: Identify Base and Values

The given expression is \( \log_5(25 \cdot 125) \). Here, the base of the logarithm is 5, and the two numbers whose product is inside the log function are 25 and 125.

Step 3: Apply Logarithm Product Rule

According to the product property of logarithms, we can rewrite \( \log_5(25 \cdot 125) \) using the property as \( \log_5 25 + \log_5 125 \). This is achieved by splitting the product inside the logarithm into two separate logarithmic terms with the same base.

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a fundamental rule that makes working with logarithms much easier. This property is especially useful when dealing with complex expressions because it simplifies calculations considerably. The rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In mathematical terms, if you're given two numbers, let's say \( a \) and \( c \), the property can be expressed as:\[\log_b(a \, \cdot \, c) = \log_b a + \log_b c.\]Here, \( b \) is the base of the logarithm. This property is particularly useful in breaking down complicated logarithmic expressions into simpler parts.

• Allows easier computation and understanding of logarithms.
• Helps in solving equations involving logarithms by simplifying expressions.

In educational exercises, such as our current example of \( \log_5(25 \, \cdot \, 125) \), we use this property to separate the expression into manageable pieces. This would look like \( \log_5 25 + \log_5 125 \), which is much simpler to handle.

Educational Exercises

Educational exercises often involve scenarios that help understand mathematical rules and properties. These exercises are designed to aid in mastering concepts through practical application. They usually involve:

• Breaking down complicated problems into more straightforward steps.
• Providing real-world or abstract examples to apply the rules.
• Giving immediate practice opportunities to reinforce the learned concepts.

In the given exercise, students are encouraged to utilize the product rule of logarithms. By applying this, they can practically see how the property makes dealing with logarithmic expressions easier. The exercise asks to express \( \log_5(25 \, \cdot \, 125) \) as a sum. By doing so, they reinforce understanding of how multiplying inside a logarithm affects the log operation itself.

Algebraic Expressions

Logarithms frequently appear within algebraic expressions, and knowing how to manipulate these expressions is an essential skill. Algebraic expressions can include terms with variables, constants, operations, and functions like logarithms. When logarithms are part of the expression, understanding their properties can:

• Help simplify complex expressions.
• Assist in solving equations involving logarithmic terms.
• Optimize the solving process by rearranging terms to facilitate calculation.

Consider the exercise from before: \( \log_5(25 \, \cdot \, 125) \). Here, the product is inside a log, so using the product rule allows for simplification into two separate, more manageable logarithmic terms — \( \log_5 25 + \log_5 125 \). Such manipulations make it easier to integrate these expressions into more extensive algebraic solutions, thereby improving efficiency and understanding in solving complex equations.