Problem 84 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to condense the expression.$$2[\ln x-\ln (x+1)-\ln (x-1)]$$.

Step-by-Step Solution

Answer

The condensed form of the given expression is \(\ln[(x^2)/((x+1)^2*(x-1)^2)]\).

Step 1: Distribute the 2 into the values within the brackets

This changes the expression to \(2\ln x - 2\ln(x+1) - 2\ln(x-1)\).

Step 2: Apply the reverse of the product and quotient properties of logarithms

Since we have subtraction, which in the logarithmic rules corresponds to division, this changes our expression to \(\ln(x^2) - \ln[(x+1)^2] - \ln[(x-1)^2]\). Here, squaring was applied to the entire argument so 2 could be moved in front of the natural logarithm.

Step 3: Combine the expressions

By continuing to apply the quotient rule of logarithms in a reverse, the expression condenses to \(\ln[(x^2)/((x+1)^2*(x-1)^2)]\). This is now a single logarithm, effectively condensing the original logarithmic expression.

Key Concepts

Condensing Logarithms

Condensing logarithms is a useful technique to simplify an expression involving multiple logarithmic terms into a single logarithm. This process frequently employs various properties of logarithms, such as the product and quotient rules. In general, the objective of condensing is to rewrite a complicated expression containing multiple logarithms as a single logarithmic term for easier interpretation or computation.
To condense, we often leverage the following rules:

• The Product Rule: \ \(\log_b(M) + \log_b(N) = \log_b(M \cdot N) \)
• The Quotient Rule: \ \(\log_b(M) - \log_b(N) = \log_b(M/N) \)
• The Power Rule: \ \(a \cdot \log_b(M) = \log_b(M^a) \)

By rearranging and applying these properties, you can simplify complex logarithmic expressions efficiently, as was shown in the exercise where the expression was condensed to a single logarithmic form.

Quotient Rule

The quotient rule for logarithms is a fundamental property that allows us to simplify expressions with logarithmic terms involving division. This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically, it is expressed as:

• \ \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)

In the context of condensing logarithms, the quotient rule is applied in reverse. For instance, if you have a series of logarithmic terms being subtracted, as seen in the step-by-step solution, you can condense them into a single logarithmic expression by representing them as a quotient.
Utilizing the quotient rule helps transform lengthy expressions into simpler forms, thus augmenting their manageability and clarity.

Logarithmic Expressions

Logarithmic expressions can come in various forms and often involve multiple terms and operations. These expressions make use of the properties of logarithms—encompassing addition, subtraction, and exponentiation—to be rewritten or evaluated in different ways.
Logarithmic expressions are useful in solving equations involving exponential growth or decay, and in signal processing or other fields dealing with multiplicative relationships.
Here's a recap of the steps for handling a logarithmic expression like the one in the exercise:

• Start by distributing any coefficients if needed, this sometimes involves using the power rule.
• Look for opportunities to apply the product or quotient rules to condense the expression into fewer logarithms.
• Combine logarithmic terms using their respective properties to simplify the expression further.

Understanding logarithmic expressions and their transformations is crucial in mathematics, as they appear frequently in both algebra and calculus.

Natural Logarithm

A natural logarithm, denoted as \(\ln\), is a specific logarithm that uses the base \(e\), where \(e\) is approximately equal to 2.71828. It holds fundamental importance in various branches of mathematics, especially calculus and complex analysis.
Natural logarithms are frequently used because they arise naturally in the process of solving exponential growth models and calculations involving the constant \(e\). Given its prevalence in real-world applications, from biology to finance, understanding how to work with \(\ln\) in various expressions is indispensable.
In the context of condensing expressions, knowing how to manipulate the natural logarithm using similar properties as common logarithms (like the product, quotient, and power rules) is a key skill. These properties make it possible to rewrite expressions involving \(\ln\) more conveniently, as demonstrated in the provided exercise.