Problem 44 - College Algebra

Question

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln x+\ln 3 $$

Step-by-Step Solution

Answer

\(\ln (3x)\)

Step 1: Identify the appropriate logarithmic property

The given expression is \(\ln x + \ln 3\). It is a sum of two logarithms which suggests to use the product rule of logarithms which states that the sum of the logs is equal to the log of the product of the numbers, \(\log_a m + \log_a n = \log_a (mn)\).

Step 2: Apply the product rule to the logarithms

Applying the product rule here, \(\ln x + \ln 3\) becomes \(\ln (x*3)\).

Step 3: Simplify the Expression

Simplify the expression, \(\ln (x*3)\) to \(\ln 3x\).

Key Concepts

Properties of Logarithms

Logarithms have several interesting properties that can simplify complex expressions. These properties are derived from the behavior of logarithmic functions and are key tools in various mathematical calculations. Understanding these properties help students handle logarithmic expressions more efficiently. Here are the primary properties:

• Product Rule: The log of a product is equal to the sum of the logs. For instance, \( \log_b(mn) = \log_b m + \log_b n \).
• Quotient Rule: The log of a quotient is the difference of the logs. It can be expressed as \( \log_b(\frac{m}{n}) = \log_b m - \log_b n \).
• Power Rule: The log of a number raised to an exponent is the exponent times the log of the number: \( \log_b(m^n) = n \cdot \log_b m \).

Recognizing when and how to apply these properties is crucial when solving logarithmic equations.

Product Rule

The Product Rule states that when you have a sum of logarithms, you can combine them into a single log. This happens when the logs have the same base and represent the log of a product.Let's say we have \( \ln a + \ln b \). According to the Product Rule, this can be simplified to \( \ln (a \cdot b) \).This property is particularly useful for simplifying expressions and solving problems more effectively. It only applies when the base of the logarithms is the same, allowing us to turn the expression involving the sum of logs into a simpler single logarithm.In our given problem, using this rule, \( \ln x + \ln 3 \) was turned into the single form \( \ln (x \cdot 3) \). This simplified approach can often help save time in calculations.

Condensing Logarithms

Condensing logarithms involves merging multiple logarithmic terms into a single term, using applicable logarithmic properties.In the context of the given exercise, where we have \( \ln x + \ln 3 \), condensing means using the Product Rule to combine them into a single logarithm: \( \ln (3x) \).The goal of condensing is to make expressions simpler and more manageable, especially when dealing with lengthy algebraic operations. By relying on properties like the Product Rule, you reduce clutter and gain a clearer understanding of the expression.

Simplifying Logarithms

Simplifying logarithmic expressions is pivotal, as it allows for easier manipulation and evaluation. A simplified expression is typically straightforward, making it much more comprehensible and easier to work with.In the case of \( \ln (x \cdot 3) \), the expression can be rewritten as \( \ln 3x \) which is a more concise representation of the original form \( \ln x + \ln 3 \).While simplification doesn't always make the numbers smaller, it often reduces complexity by lowering the number of operations you need to perform. Thus, simplification is a useful technique, aiding in clear thinking and efficient problem-solving.