Problem 105 - A Graphical Approach to College Algebra

Question

The equations are identities because they are true for all real numbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any real number. $$\frac{1}{3} \ln \left(\frac{x^{2}+1}{5}\right)-\frac{1}{3} \ln \left(\frac{x^{2}+4}{5}\right)=\ln \sqrt[3]{\frac{x^{2}+1}{x^{2}+4}}$$

Step-by-Step Solution

Answer

The identities match after simplification, confirming that they are equal for all real numbers \(x\).

Step 1: Identify the Problem

We need to show that the left-hand side of the equation \[\frac{1}{3} \ln \left(\frac{x^{2}+1}{5}\right)-\frac{1}{3} \ln\left(\frac{x^{2}+4}{5}\right)\] is equal to the right-hand side, which is \(\ln \sqrt[3]{\frac{x^{2}+1}{x^{2}+4}}\).

Step 2: Apply Logarithm Properties

Use the property \(\ln a - \ln b = \ln \frac{a}{b}\) to combine the logarithm terms on the left-hand side of the equation. It becomes:\[\frac{1}{3} \cdot \ln \left(\frac{\frac{x^{2}+1}{5}}{\frac{x^{2}+4}{5}}\right)\] which simplifies to:\[\frac{1}{3} \cdot \ln \left(\frac{x^{2}+1}{x^{2}+4}\right)\] after canceling the \(5\) in the denominator.

Step 3: Simplify Using the Power Rule for Logarithms

Apply the power rule of logarithms \(a \ln b = \ln b^a\). For our expression:\[\frac{1}{3} \ln \left(\frac{x^{2}+1}{x^{2}+4}\right)\] becomes:\[\ln \left(\frac{x^{2}+1}{x^{2}+4}\right)^{\frac{1}{3}}\] or equivalently:\[\ln \sqrt[3]{\frac{x^{2}+1}{x^{2}+4}}\].

Step 4: Conclude the Identity

Now that the left-hand side \(\ln \sqrt[3]{\frac{x^{2}+1}{x^{2}+4}}\) matches the right-hand side, we confirm that the expressions are equal for all real numbers \(x\).

Key Concepts

Properties of Logarithms

The properties of logarithms are fundamental tools in algebra and calculus, used to transform and simplify complex expressions. These properties allow us to break down logarithmic expressions into more manageable parts or to combine simple expressions into more complex ones. Here are a few essential properties:

• Product Property: \(\ln(a \cdot b) = \ln a + \ln b\)
• Quotient Property: \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\)
• Power Rule: \(a \cdot \ln b = \ln b^a\)

Understanding these properties helps to rearrange and simplify logarithmic expressions, as seen in our exercise. We leveraged the quotient property to simplify the logarithmic difference into a single logarithmic term. Additionally, the power rule was used to resolve the fraction coefficient into an exponent inside the logarithm, showcasing these properties' utility in simplifying the equation.

Simplification of Expressions

Simplification of expressions involves reducing complex expressions into simpler forms while maintaining their equivalency. The goal is to make calculations easier and provide clearer insights into the mathematical relationship.

To simplify the expression \(\frac{1}{3} \ln \left(\frac{x^{2}+1}{5}\right)-\frac{1}{3} \ln\left(\frac{x^{2}+4}{5}\right)\), we first needed to recognize each term's structure.

By applying the properties of logarithms, we combined the two separate logarithms into one using the Quotient Property:

• Combine: \(\ln a - \ln b = \ln \frac{a}{b}\)
• Result: \(\frac{1}{3} \ln \left(\frac{\frac{x^{2}+1}{5}}{\frac{x^{2}+4}{5}}\right) = \frac{1}{3} \ln \left(\frac{x^{2}+1}{x^{2}+4}\right)\)

Next, by applying the Power Rule, we expressed the entire logarithm term to the power as a single logarithm. This process was crucial in confirming the original identities were equivalent.

In essence, simplification through logarithmic properties transforms the problem into a more recognizable form.

Mathematical Proofs

Mathematical proofs confirm the equivalency of expressions or the validity of statements using logical reasoning and known facts. In this exercise, we demonstrated an identity, which is an equation true for all values in its domain.

The proof began with the given identity, where the left-hand expression involving logarithms was complex. We aimed to show it equates to the simpler right-hand expression:

• Start by using the properties of logarithms to manipulate the left-hand side.
• Simplify each component systematically while maintaining logical steps.
• Show that every transformation preserves equivalency until reaching the desired right-hand side expression.

This approach confirms the expression remains true for any real number \(x\). Thus, through these transformations and simplifications, a successful proof not only verifies the identity but enhances understanding of the underlying mathematical principles.