Problem 51 - Introductory and Intermediate Algebra for College Students 4th

Question

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. $$5 \ln x-2 \ln y$$

Step-by-Step Solution

Answer

The logarithmic expression simplifies to \(\ln \left(\frac{x^5}{y^2}\right)\)

Step 1: Apply Logarithmic Exponentiation Rule

The first step is to recognize that the expressions \(5 \ln x\) and \(2 \ln y\) can be rewritten using the rule that allows removing the coefficient of the logarithm and placing it as an exponent of the argument inside the logarithm. The expression \(5 \ln x-2 \ln y\) then transforms into \(\ln x^5 - \ln y^2\).

Step 2: Apply Logarithmic Subtraction Rule

Next, you will use the rule that states that subtraction outside the log translates into division inside the log. The expression \(\ln x^5 - \ln y^2\) then simplifies to \(\ln \left(\frac{x^5}{y^2}\right)\)

Key Concepts

Properties of Logarithms

Logarithms have specific properties that make complex expressions easier to work with. These properties allow us to manipulate and simplify logarithmic expressions, turning them into more manageable forms. Some of the core properties of logarithms include:

• The Product Rule: \(\log_b (MN) = \log_b M + \log_b N\)
• The Quotient Rule: \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\)
• The Power Rule: \(\log_b (M^n) = n \log_b M\)

These rules are powerful tools that can be used to both expand and condense logarithmic expressions. By understanding and applying these properties, you can convert cumbersome logs into simple expressions that are easier to evaluate or compare.

Logarithmic Exponentiation Rule

The Logarithmic Exponentiation Rule is fundamental in transforming logarithmic expressions. This rule states that if you have a logarithm with a coefficient in front, you can move the coefficient as the power of the logarithm's argument. For example, the expression \(c \ln a\) can be rewritten as \(\ln a^c\).
This transformation is useful for simplifying the appearance and complexity of expressions, particularly when combining or simplifying. By using the exponentiation rule, you can transform expressions like \(5 \ln x\) into \(\ln x^5\), reducing the coefficient and adding clarity.

Logarithmic Subtraction Rule

The Logarithmic Subtraction Rule is another essential tool in your logarithm toolkit. This rule holds that when you subtract one logarithm from another, it is equivalent to taking the logarithm of a quotient. In simpler terms:

• \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\)

This property is vital for condensing multiple logarithms into a single, streamlined expression. In practice, it allows you to simplify expressions like \(\ln x^5 - \ln y^2\) to \(\ln \left(\frac{x^5}{y^2}\right)\).
Dealing with a single logarithm instead of multiple simplifies computation and makes evaluating expressions much easier.

Condensing Logarithms

Condensing logarithms is the process of rewriting multiple logarithmic expressions into a single, unified logarithm. This process often involves the previously mentioned properties like the exponentiation and subtraction rules.
In the example \(5 \ln x - 2 \ln y\), we used exponentiation to move the coefficients as exponents and subtraction to create a division inside the logarithm. This results in the simplified expression \(\ln \left(\frac{x^5}{y^2}\right)\).

• Condensed expressions are more elegant and often easier to handle.
• They reduce complexity and enhance readability.

Grasping how to condense logarithms efficiently allows one to swiftly solve or evaluate expressions that might initially seem daunting.