Problem 252 - Precalculus

Question

For the following exercises, expand each logarititim as much as possible. Rewite each expression as a sum, difference, or product of logs. $$ \ln (3 a b \cdot 5 c) $$

Step-by-Step Solution

Answer

\( \ln(3) + \ln(a) + \ln(b) + \ln(5) + \ln(c) \)

Step 1: Understand the Expression

The given expression is \( \ln (3ab \cdot 5c) \). The objective is to rewrite this logarithmic expression as a sum, difference, or product of logs using logarithmic properties.

Step 2: Apply the Logarithm of a Product Property

Recall the property that \( \ln(xy) = \ln(x) + \ln(y) \). Apply this to \( \ln(3ab \cdot 5c) \) to get \( \ln(3ab) + \ln(5c) \).

Step 3: Expand Each Component Using Properties

We'll further expand each logarithmic term. For \( \ln(3ab) \), apply the product rule again to obtain \( \ln(3) + \ln(a) + \ln(b) \). Similarly, for \( \ln(5c) \), apply the product rule to get \( \ln(5) + \ln(c) \).

Step 4: Combine All Logarithmic Terms

Combine the expressions from the previous step: \( \ln(3) + \ln(a) + \ln(b) + \ln(5) + \ln(c) \). This is the expanded form of the original expression as a sum of logs.

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools that make working with logarithms easier. They allow us to simplify complex logarithmic expressions into manageable parts. Understanding these properties helps in expanding or condensing logarithmic expressions:

• Product Rule: This states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, \( \ln(xy) = \ln(x) + \ln(y) \), can be applied to break down a multiplication inside the logarithm into an addition outside.
• Quotient Rule: This property states that the logarithm of a quotient is the difference of the logarithms: \( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \).
• Power Rule:** This rule indicates that the logarithm of a power is the exponent times the logarithm of the base: \( \ln(x^n) = n \cdot \ln(x) \).

By applying these properties, especially the product rule, you can expand logarithms more easily, as seen in the exercise. The goal is to transform a single complex logarithmic expression into simpler parts that are summed or subtracted, facilitating better understanding and manipulation.

Product Rule for Logarithms

The product rule is incredibly useful when working with logarithms, especially for expansion. This rule allows us to express the logarithm of a product as a sum of logarithms. For example:

• In the expression \( \ln(3ab \cdot 5c) \), it may initially look daunting, but the product rule breaks it down to: \[ \ln(3ab) + \ln(5c) \].
• Then, using the product rule again, separate terms further: \[ \ln(3) + \ln(a) + \ln(b) + \ln(5) + \ln(c) \].

This step-by-step application of the product rule clearly demonstrates how multiplication within logarithms converts to addition outside. Each factor within the logarithmic expression becomes a separate term, making complex expressions less intimidating. The product rule is particularly handy when dealing with variables and numbers, enabling a structured and logical breakdown of expressions.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a specific type of logarithm with a base \(e\), where \(e\) is approximately equal to 2.718. It is widely used in various scientific and mathematical applications due to its unique properties.

• Natural Base: The constant \(e\) is the base of natural logarithms, and it arises naturally in various growth processes, including compound interest calculations and population growth models.


• Simplifying Logarithmic Expressions: The rules for manipulating \(\ln\) expressions are the same as those for other logarithmic bases, such as the product, quotient, and power rules mentioned earlier.


• Applications: Natural logarithms are present in fields like calculus, where they facilitate integration and differentiation. Given their base \(e\), they often simplify complex equations that model real-world phenomena.

In exercises like the one we're discussing, using the natural logarithm provides consistency and utility across various operations. Understanding \(\ln\) ensures that you can efficiently manipulate expressions and solve problems related to growth and decay.