Problem 6 - Algebra A Combined Function

Question

Write each sum as a single logarithm. Assume that variables represent positive numbers. $$ \log _{5} y^{3}+\log _{5}(y-7) $$

Step-by-Step Solution

Answer

\( \log_5 (y^3(y-7)) \)

Step 1: Identify the Properties

We will apply the properties of logarithms. Specifically, we need the Product Property of Logarithms, which states that \( \log_b A + \log_b B = \log_b (A \cdot B) \).

Step 2: Apply the Product Property

Using the Product Property to combine the logarithms into a single logarithm: \( \log_5 y^3 + \log_5 (y-7) = \log_5 (y^3 \cdot (y-7)) \).

Step 3: Express the Final Single Logarithm

The expression \( \log_5 (y^3 \cdot (y-7)) \) is the combined logarithm. Thus, the sum is written as a single logarithm: \( \log_5 (y^3(y-7)) \).

Key Concepts

Product Property of Logarithms

Understanding the product property of logarithms is crucial for manipulating and simplifying logarithmic expressions. This property tells us how to combine two logarithms into one. If you have two logarithms with the same base, you can use this property.

This is what the product property states:

• Given two logarithms with the same base, \( \log_b A \) and \( \log_b B \), you can combine them as \( \log_b A + \log_b B = \log_b (A \cdot B) \).

The product property essentially allows you to "combine" the numbers that the logarithms "point to" by multiplying them.
It’s a foundational tool when simplifying expressions, as seen in the exercise where \( \log_5 y^3 + \log_5 (y-7) \) are combined into \( \log_5 (y^3(y-7)) \). This makes calculations easier and expressions more straightforward to handle by reducing the number of terms.

Combining Logarithms

Combining logarithms involves the process of using logarithmic properties, such as the product property, to merge multiple logarithmic terms into a single, often simpler, expression.

Why do we combine logarithms? It’s mainly for simplification and to make expressions easy to work with or solve.

• The process involves identifying common bases and applying properties of logarithms.
• Combining makes lengthy expressions more compact.

As with the given exercise, by applying the product property, separate logarithms with the same base \( \log_5 y^3 + \log_5 (y-7) \) were combined into a single, much clearer expression: \( \log_5 (y^3(y-7)) \).
This reduction is particularly helpful in solving equations or comparing logarithmic expressions.

Properties of Logarithms

Logarithms have several crucial properties that help in manipulating and transforming logarithmic expressions. Three main properties interact to simplify expressions:

• Product Property: as previously explained, this combines sums of logarithms into a product.
• Quotient Property: \( \log_b A - \log_b B = \log_b \left(\frac{A}{B}\right) \). It helps in dealing with differences of logarithms.
• Power Property: \( \log_b (A^c) = c \cdot \log_b A \). It allows moving exponents outside of the logarithm.

These properties are the cornerstone for rewriting, reducing, and solving logarithmic expressions.

In any exercise dealing with logarithms, recognizing which property to apply is key to simplifying and working through the problem efficiently. For the given exercise specifically, using the product property was the most direct route to simplify and solve the expression.