Problem 52 - 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. $$ \frac{1}{3} \ln x+\ln y $$

Step-by-Step Solution

Answer

The given expression \(\frac{1}{3} \ln x + \ln y\) condensed into a single logarithm whose coefficient is 1 is: \(\ln (x^{1/3}y)\)

Step 1: Apply the power rule of logarithms

We will first apply the power rule of logarithms for the term \(\frac{1}{3} \ln x\). According to the power rule, \(a \ln b = \ln (b^a)\). So the given expression becomes: \(\ln (x^{1/3}) + \ln y\)

Step 2: Apply the product rule of logarithms

Now we will apply the product rule of logarithms to combine the two logarithms into one. According to the product rule, \(\ln a + \ln b = \ln (a \times b)\). So, \(\ln (x^{1/3}) + \ln y = \ln ((x^{1/3}) y)\)

Step 3: Rewrite in correct form

Finally, we rewrite the expression as a single logarithm whose coefficient is 1. So the final expression is: \(\ln (x^{1/3}y)\)

Key Concepts

Properties of Logarithms

Logarithms have certain properties that make them incredibly useful for simplifying and manipulating mathematical expressions. Understanding these properties is key to working effectively with logarithmic expressions. Here are a few essential properties you should know:

• Product Property: \(\ln a + \ln b = \ln (a \times b)\). This property allows you to combine the logarithms of a product into a single logarithm.
• Quotient Property: \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\). This helps in expressing the logarithm of a division.
• Power Property: \(a \ln b = \ln (b^a)\). This is useful when the logarithm is initially multiplied by a factor.

Recognizing and using these properties can greatly simplify the process of solving logarithmic equations and expressions by reducing them to a more manageable form.

Power Rule of Logarithms

The power rule of logarithms is a special property that makes it easy to handle exponents within logarithmic expressions. According to this rule, if you have a multiplication factor in front of a log, you can rewrite it as an exponent inside the log. So for the expression \(a \ln b\), it becomes \(\ln (b^a)\).

This is highly beneficial especially when dealing with fractional coefficients as exponents. For instance, in the exercise, we converted \(\frac{1}{3} \ln x\) to \(\ln (x^{1/3})\), which translates to finding the cube root of x inside the logarithm. This process simplifies the expression by internally moving the factor, making the expression easier to evaluate or further manipulate.

Using the power rule effectively requires some practice, but once mastered, it becomes a valuable tool in simplifying logarithmic expressions that involve constants or fractions.

Product Rule of Logarithms

The product rule for logarithms extremely simplifies the handling of sums of logarithms. The rule states that the sum of two logarithms with the same base can be condensed into a single logarithm of the product of their arguments. Mathematically, this is expressed as \(\ln a + \ln b = \ln (a \times b)\).

In the given exercise, after rewriting \(\frac{1}{3} \ln x\) as \(\ln (x^{1/3})\) using the power rule, we then applied the product rule. This enabled us to combine the expression with \(\ln y\) into \(\ln ((x^{1/3}) y)\).

This property is particularly useful when you're tasked with "condensing" logarithmic expressions, which means turning multiple log terms into one. This simplification is crucial in solving complex logarithmic equations as it eases the process of solving like terms or isolating variables.

Condensing Logarithms

The process of condensing logarithms involves using various properties of logarithms to rewrite multiple logarithmic terms into a single, simplified form. The aim is to reduce expressions to one logarithm, typically with a coefficient of 1 when possible.

When condensing logarithms, you apply the properties discussed so far: Power Rule to bring multipliers inside the log, Product Rule for combining logs added together, and the Quotient Property for logs that are subtracted. This can make evaluating expressions by hand simpler and clearer.

In our exercise, we applied these properties consecutively. Using the power rule, \(\frac{1}{3} \ln x\) was rewritten as \(\ln (x^{1/3})\). Then, using the product rule, we combined the entire expression into a single logarithm \(\ln (x^{1/3} y)\). This condensed form is neater and often more useful in further calculus applications.

Condensing is a powerful technique for algebraic manipulation and helps in solving logarithmic equations efficiently.