Problem 52 - 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. $$7 \ln x-3 \ln y$$

Step-by-Step Solution

Answer

The condensed form of the given logarithmic expression is \( \ln \frac{x^7}{y^3} \).

Step 1: Apply the Power Rule

The power rule of logarithm can be applied in the given expression. This rule allows us to move the coefficient of a logarithm to the exponent of its argument. Apply the power rule to both \(7 \ln x\) and \(-3 \ln y\) to get \( \ln x^7 - \ln y^3 \).

Step 2: Apply the Quotient Rule

Once the expression has been simplified using the power rule, the quotient rule can be applied. The quotient rule combines two subtracted logarithmic expressions into one by constructing a quotient of the arguments inside one logarithm. Thus, \( \ln x^7 - \ln y^3\) becomes \( \ln \frac{x^7}{y^3} \).

Step 3: Final solution

The expression has been condensed into a single logarithm with a coefficient of 1, so the final solution is \( \ln \frac{x^7}{y^3} \). If the variables x and y had specific values, we would substitute those in at this point and compute the expression.

Key Concepts

Properties of Logarithms

Logarithms come with a set of rules that help us manipulate and simplify expressions. Understanding these properties is crucial to working with logarithmic problems effectively. These rules allow us to expand or condense logarithmic expressions based on different scenarios involving multiplication, division, and exponentiation of their arguments.

• **Product Rule**: This rule states that the logarithm of a product can be expressed as the sum of the logarithms of its factors. Mathematically, this is shown as: \[\log_b(MN) = \log_b(M) + \log_b(N) \]
• **Quotient Rule**: Similarly, the logarithm of a quotient can be written as the difference between the logarithms of the numerator and denominator:\[\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\]
• **Power Rule**: The logarithm of a power can be simplified by bringing the exponent in front of the logarithm:\[\log_b(M^n) = n\log_b(M) \]

These properties are immensely helpful as they turn complex logarithmic calculations into simpler arithmetic ones. In the given exercise, we apply these properties to transform a sum and difference of logarithms into a single, streamlined expression.

Power Rule

The Power Rule is a significant property of logarithms that allows us to manipulate expressions with exponents efficiently. According to this rule, when you have a logarithm of something raised to a power, you can pull the exponent out in front of the logarithm. This turns a power inside the logarithm into a multiplier outside. In terms of a mathematical formula, the Power Rule can be expressed as:\[\log_b(M^n) = n \cdot \log_b(M)\]Let's illustrate this with the exercise problem:

• The expression given is: \[7 \ln(x)\]Using the Power Rule, this becomes:\[\ln(x^7)\]
• Similarly, for the expression \[-3 \ln(y)\]applying the Power Rule results in:\[-\ln(y^3)\]

By using the Power Rule, we have successfully transferred the coefficients of the logarithms into exponents within the logarithm. Doing so simplifies the expressions further and prepares them for the application of other logarithmic properties.

Quotient Rule

The Quotient Rule is particularly useful when working with logarithmic expressions involving division. This rule tells us that the logarithm of a division is equal to the difference between the logarithm of the numerator minus the logarithm of the denominator. According to the Quotient Rule, the property can be summarized as:\[\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\]Looking at our exercise, after applying the Power Rule, we had:\[\ln(x^7) - \ln(y^3)\]Using the Quotient Rule, these two logarithmic terms can be combined into one:\[\ln\left(\frac{x^7}{y^3}\right)\]This transformation is powerful because it collapses a subtraction into a single logarithmic expression. This progression from two separate logarithmic terms to one makes the process more streamlined and shows how different properties of logarithms work together to simplify expressions.