Problem 38 - Precalculus

Question

Write each expression as a single logarithm. $$2 \log _{b} u+3 \log _{b} v$$

Step-by-Step Solution

Answer

The expression is \( \log_b(u^2 \, v^3) \).

Step 1: Use Product Property of Logarithms

Recall the product property of logarithms, which states that \( a \log_b(x) + c \log_b(y) = \log_b(x^a \, y^c) \). We aim to combine the logs using this property.

Step 2: Apply the Product Property

Apply the property to the given expression \( 2 \log_b(u) + 3 \log_b(v) \). This can be rewritten as \( \log_b(u^2) + \log_b(v^3) \).

Step 3: Combine the Logarithms

Apply the product rule to combine the two logarithms: \( \log_b(u^2) + \log_b(v^3) = \log_b(u^2 \cdot v^3) \).

Step 4: Write Final Expression

The expression is now a single logarithm: \( \log_b(u^2 \, v^3) \). This is the simplest form of the expression as a single logarithm.

Key Concepts

Product Property of Logarithms

The product property of logarithms is a vital tool for simplifying logarithmic expressions. This property states: \( a \log_b(x) + c \log_b(y) = \log_b(x^a \cdot y^c) \). It's like condensing multiple logarithmic terms into a single, cohesive term.

This property is useful because it allows us to handle multiple terms more efficiently, especially when dealing with exponential growth or decay problems. Imagine if we want to simplify expressions with repeated bases raised to different powers.

• Instead of dealing with each term separately, as shown in the formula, you multiply the bases after raising them to their respective powers.
• This simplifies calculations, especially in more complex algebraic problems.

Identifying when to use this property can make logarithmic equations smoother and more straightforward.

Combining Logarithms

Combining logarithms is about simplifying expressions where you have multiple log terms with the same base. The ultimate goal is to end with a single logarithmic expression. The product property helps, but knowing when and how to combine them is key.

Consider the expression \( \log_b(u^2) + \log_b(v^3) \). Combining these logs means you use the property: if you have \( \log_b(a) + \log_b(b) \), it becomes \( \log_b(a \cdot b) \).

• This means taking the logarithm of the product of two numbers is equivalent to the sum of their logarithms.
• This technique is not only cleaner but can also simplify expressions for other calculations or interpretations.

By recognizing these patterns, you're empowering yourself to handle more advanced logarithmic manipulations efficiently.

Logarithmic Functions

Logarithmic functions are an essential part of algebra and calculus. They are the inverse operations of exponential functions. Understanding them is crucial for solving equations involving growth and decay.

A logarithmic function is typically written as \( \log_b(x) \), where \( b \) is the base. It's asking: "To what power should \( b \) be raised to give \( x \)?"

• Logarithms translate multiplication into addition, allowing for easier manipulation of mathematical expressions.
• They have applications not only in pure mathematics but also in fields like science, engineering, and computer science.

By understanding the behavior and properties of logarithmic functions, you can better appreciate how they model real-world phenomena such as sound intensity, pH levels, and even social network connections.