Problem 53 - 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. $$3 \ln x-\frac{1}{3} \ln y$$

Step-by-Step Solution

Answer

The simplified expression for \(3 \ln x - \frac{1}{3} \ln y\) when written as a single logarithm with coefficient 1 is \(\ln\left(\frac{x^3} {y^{\frac{1}{3}}}\right)\)

Step 1: Apply the Power Rule

Firstly, the expression is \(3 \ln x - \frac{1}{3} \ln y\). The power rule of logarithms states that \(a\ln m = \ln (m^a)\). Applying this to the expression, it simplifies to \(\ln (x^3) - \ln (y^{\frac{1}{3}})\)

Step 2: Apply the Quotient Rule

The quotient rule of logarithms states that \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\). By applying the quotient rule, the expression further simplifies to \(\ln\left(\frac{x^3} {y^{\frac{1}{3}}}\right)\)

Step 3: Convert to a Single Logarithm

Finally, the expression becomes a single logarithm with the coefficient of 1, as \(\ln\left(\frac{x^3} {y^{\frac{1}{3}}}\right)\). This is the simplification of the given expression using properties of logarithms.

Key Concepts

Condensing Logarithms

Imagine you're working with multiple logarithmic expressions, and your goal is to combine them into a single, simplified logarithm. This process is known as condensing logarithms. It's like taking many pieces of a puzzle and combining them to see the whole picture clearly. To do this effectively, you need to understand and apply the properties of logarithms, such as the product, quotient, and power rules.

When you are dealing with expressions like 3 ln x - (1/3) ln y, you must first use the power rule to rewrite each term in a form that allows for condensation. Only when each logarithm is in its simplest form can you then proceed to combine them into a single expression. Remember that your ultimate goal is to have a single logarithm with a coefficient of 1. This not only helps in simplifying complex expressions but also is often required for solving higher-level mathematics problems.

Power Rule of Logarithms

The power rule of logarithms is a fundamental tool that allows us to simplify logarithmic expressions where the argument (the number inside the logarithm) is raised to a power. Think of it as being able to take the exponent and pull it in front as a multiplier to the logarithm, essentially 'unpacking' the power from within the log function.

For example, when you have an expression like 3 ln x, according to the power rule, this can be rewritten as \( \ln(x^3) \). This rule is based on the principle that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number. It's a simple yet powerful property that can be used to manipulate and condense expressions. Applying the power rule can transform a seemingly complex logarithm into a more manageable form, making it easier to work with, especially when dealing with equations or comparing different logarithmic terms.

Quotient Rule of Logarithms

When it comes to dividing logarithms, the quotient rule of logarithms is your best friend. This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. To put it more concretely, \( \ln\frac{a}{b} = \ln a - \ln b \).

So, after you've used the power rule, and you have two logarithms subtracted from each other, like \(\ln (x^3) - \ln (y^{\frac{1}{3}})\), the quotient rule lets you combine these logs into a single expression. In the context of our original problem, applying the quotient rule turns \(\ln (x^3) - \ln (y^{\frac{1}{3}})\) into \(\ln\left(\frac{x^3}{y^{\frac{1}{3}}}\right)\), which is much simpler to handle. This step is crucial for condensation as it reduces the expression to a sole logarithm that describes the ratio between the two original terms' arguments. The quotient rule is particularly valuable when you're solving equations with logarithms in them or when you're working with growth and decay problems in logarithmic form.