Problem 37 - Elementary and Intermediate Algebra

Question

Write logarithm as a sum. Then simplify, if possible. \(\log 5 x y z\)

Step-by-Step Solution

Answer

The expression \( \log 5xyyz \) simplifies to \( \log 5 + \log x + \log y + \log z \).

Step 1: Apply the Product Rule of Logarithms

The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the factors. Specifically, for any positive numbers \(a\), \(b\), and \(c\), \( \log(abc) = \log a + \log b + \log c \). Apply this to the expression \( \log 5xyy \) to get \( \log 5 + \log x + \log y + \log z \).

Step 2: Check for Simplification

Look at each term in the expression \( \log 5 + \log x + \log y + \log z \) to determine if there is any further simplification possible. Each term is in its simplest form as there are no like terms or products of terms that can be combined further.

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a powerful tool for breaking down complex expressions into simpler pieces. It describes how you can separate the logarithm of a product into a sum of logarithms. In mathematical terms, it can be expressed as:

• \( \log(a \cdot b \cdot c) = \log a + \log b + \log c \)

This rule is extremely useful for computations and transformations involving logarithms. In our exercise, you encounter the expression \( \log(5xyz) \), which simplifies to \( \log 5 + \log x + \log y + \log z \). By using this rule, a single expression involving multiple variables becomes a series of simpler terms. This transformation makes handling logarithmic expressions much easier. Every factor in the product gets its logarithmic term, really helping with clarity and computation!

Simplification

When simplifying logarithmic expressions, the goal is to present them in the most straightforward form. Simplification often involves looking for patterns or terms that can be combined or reduced. Once you've used the product rule to expand the expression, you should check each term for potential simplification opportunities.

• Look for like terms or common bases.
• See if there are any logarithmic identities applicable.

In the exercise provided, the expression \( \log 5 + \log x + \log y + \log z \) is already in its simplest form. There are no like terms or further combination possibilities. It is important to always perform this check to ensure that the expression cannot be simplified further, ensuring accuracy and precision in your work.

Sum of Logarithms

The sum of logarithms represents a scenario where multiple logarithmic terms are added together. It's crucial because it allows for the breakdown of complex logarithmic expressions into manageable parts. This effectively maps the logarithm of a product to a more easily handled sum.

• The sum \( \log 5 + \log x + \log y + \log z \) is derived from the product \( \log(5xyz) \).
• Each addend represents one component of the product within the logarithm.

Understanding how to write the sum of logarithms is essential for solving and simplifying logarithmic equations, especially in algebra and calculus. It's important to remember that each component of the initial product gets translated into its own logarithmic term in the sum, providing a clear and organized representation of the components involved.