Problem 5 - Intermediate Algebra

Question

Write each sum as a single logarithm. Assume that variables represent positive numbers. See Example 1 . $$ \log _{6} x+\log _{6}(x+1) $$

Step-by-Step Solution

Answer

\( \log_6 (x \cdot (x+1)) \)

Step 1: Review Logarithm Properties

The sum of two logarithms with the same base can be condensed into a single logarithm by using the product property: \( \log_b a + \log_b b = \log_b (a \cdot b) \). This means we can combine the two logs by multiplying their arguments.

Step 2: Apply the Product Property

Apply the product property to the given logarithms: \( \log_6 x + \log_6 (x+1) = \log_6 (x \cdot (x+1)) \). Therefore, the sum of these logarithms is equivalent to a single logarithm of the product of their arguments.

Key Concepts

Product Property of Logarithms

The product property of logarithms is a fundamental concept that simplifies working with logarithmic expressions. When you encounter two logarithms with the same base that are added together, such as \( \log_b a + \log_b b \), you can combine them into a single logarithm. This involves multiplying the arguments of the two logarithms. By using the product property, you can express the sum as \( \log_b (a \cdot b) \).
This property is particularly useful because it condenses the expression, making calculations and further operations easier. Remember: this property only works if the bases are the same. Always check that your logarithms share a common base before applying the product property.

Combining Logarithms

Combining logarithms involves using logarithmic properties to merge multiple logarithmic terms into a single term. Using the product property is one way to do this. If you have expressions like \( \log_c a \) and \( \log_c b \), and you want one log expression, you look at their sum: \( \log_c a + \log_c b \).

• Make sure the bases \( c \) are the same.
• Multiply the arguments \( a \) and \( b \) because of the product property.
• Combine them: \( \log_c (a \cdot b) \).

It's an efficient strategy for streamlining complex algebraic processes. Not only does it simplify expressions, but it also prepares them for more advanced manipulations, such as differentiation or integration.

Single Logarithm Expression

A single logarithm expression is a condensed form of a previously more complex logarithmic equation. By reducing multiple logarithms into one, using properties like the product property, you achieve a streamlined form. For instance, in the given example of \( \log_6 x + \log_6 (x+1) \), it becomes \( \log_6 (x \cdot (x+1)) \).
Why is this important?

• It simplifies calculations, reducing errors when evaluating or solving equations.
• A single logarithm expression is easier to manage when solving real-world problems, optimizing calculations in scientific and engineering contexts.

In essence, mastering the art of condensing into a single logarithm is not only about simplification but also about enhancing accuracy and efficiency in mathematical problem-solving.