Problem 52 - College Algebra Essentials

Question

Use properties of logarithms to condense 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

\(\ln (x^{1/3}y)\)

Step 1: Applying the Power Rule

The power rule of logarithms states that \(b \log a^n = \log a^{bn}\). Apply this rule to the expression, using the fraction in front of the first log as the power for \(\ln x\). This results in: \(\ln x^{1/3} + \ln y\)

Step 2: Applying the Product Rule

The product rule of logarithms states that \(\log ab = \log a + \log b\). Here, use this rule to combine the two logarithms into one. This results in: \(\ln (x^{1/3}y)\)

Key Concepts

Logarithm Condensation

Logarithm condensation is a technique that allows us to combine multiple logarithmic expressions into a single expression. This process not only simplifies the appearance of mathematical expressions but also makes it easier to solve complex logarithmic equations. One common scenario, as seen in our original exercise, is condensing expressions where the individual logarithms are added or subtracted.

Upon simplification, the goal is to obtain a single logarithmic expression with a coefficient of 1, which indicates that it cannot be further simplified or condensed. The simplification process may involve using properties of logarithms such as the power rule and the product rule, which will be explained in the following sections.

Power Rule of Logarithms

The power rule of logarithms is an essential property that helps in the process of logarithm condensation. This rule states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. Mathematically, it's expressed as \( b \log a^n = \log a^{bn} \).

In the context of our exercise, applying the power rule transforms \( \frac{1}{3} \ln x \) into \( \ln x^{1/3} \). The coefficient outside the logarithm becomes the exponent of the argument within the logarithm, allowing further simplification with other logarithmic terms. Understanding and correctly applying the power rule can turn a complex logarithmic expression into a more manageable form.

Product Rule of Logarithms

Following the application of the power rule, the product rule of logarithms becomes the next step in the condensation process. This rule allows us to multiply arguments of logarithms directly, provided that the logarithms have the same base and are added together. The rule is formally stated as \( \log ab = \log a + \log b \).

In our exercise, the product rule is applied to combine \( \ln x^{1/3} \) and \( \ln y \) into a single logarithmic expression \( \ln (x^{1/3}y) \). When applied correctly, the product rule significantly simplifies the equation by condensing multiple logarithmic terms into one, making it easier to evaluate and manipulate further.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a type of logarithm with the base \( e \), where \( e \) is an important mathematical constant approximately equal to 2.71828. Natural logarithms are frequently used in various fields such as science, engineering, and economics due to their intrinsic appearance in growth models, compound interest calculations, and the natural behavior of complex systems.

In the given problem, \( \ln \) is used to denote the natural logarithm. It is important to recognize the natural logarithm function versus logarithms with other bases, as they are notated and behave differently, especially when considering change of base formulas. Grasping this distinction is crucial for solving more advanced problems involving exponential and logarithmic functions.