Problem 45 - Precalculus

Question

Write each expression as a single logarithm. $$\ln (x+1)+\ln (x-1)-2 \ln \left(x^{2}+3\right)$$

Step-by-Step Solution

Answer

\( \ln \left( \frac{x^2 - 1}{(x^2 + 3)^2} \right) \)

Step 1: Identify the Given Logarithmic Terms

The expression given is \( \ln (x+1)+\ln (x-1)-2 \ln \, (x^{2}+3) \). Notice that the expression involves multiple natural logarithms.

Step 2: Apply the Product Rule for Logarithms

The product rule for logarithms states that \( \ln a + \ln b = \ln (a \cdot b) \). We apply this to the first two terms: \( \ln (x+1) + \ln (x-1) = \ln ((x+1)(x-1)) \).

Step 3: Simplify Using Difference of Squares

Recognize that \((x+1)(x-1)\) is a difference of squares, which simplifies as \(x^2 - 1\). So, \(\ln ((x+1)(x-1))\) becomes \(\ln (x^2 - 1)\).

Step 4: Apply the Power Rule for Logarithms

The power rule states \( n \ln a = \ln (a^n) \). Apply this to the term \(-2 \ln (x^2 + 3)\) to get \(\ln ((x^2 + 3)^{-2})\).

Step 5: Combine All Logarithms Using the Quotient Rule

The quotient rule for logarithms states \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). Combine our simplified logarithm terms: \( \ln (x^2 - 1) - \ln ((x^2 + 3)^2) = \ln \left( \frac{x^2 - 1}{(x^2 + 3)^2} \right) \).

Key Concepts

Product Rule for Logarithms

When working with logarithms, one of the key properties to understand is the product rule. This rule states that when you add two logarithms with the same base, it's equivalent to taking the logarithm of the product of the two arguments. In mathematical terms, this is expressed as: \( \ln a + \ln b = \ln (a \cdot b) \).

Let's break this down to make it easier to grasp:

• If you have \(\ln(x+1) + \ln(x-1)\), you combine them by multiplying \((x+1)(x-1)\).
• This multiplication simplifies the expression inside the logarithm: \(x^2 - 1\) due to the difference of squares formula.

By using the product rule, you condense the expression to a single logarithm, making calculations and manipulations of the logarithms simpler.

Power Rule for Logarithms

Another critical property of logarithms is the power rule, which deals with exponents inside logarithms. The power rule states that: \( n \ln a = \ln (a^n) \).

This rule emphasizes how an exponent inside a logarithm can be moved in or out as a multiplier. Consider the following example:

• If you start with \(-2 \ln(x^2 + 3)\), you can rewrite it as \(\ln((x^2 + 3)^{-2})\) by moving the \(-2\) as an exponent inside the logarithm.
• This transformation helps to simplify expressions, particularly when you need to combine multiple logarithms.

Understanding the power rule allows you to manage exponents effectively and is a crucial tool for simplifying complex logarithmic expressions.

Quotient Rule for Logarithms

The quotient rule for logarithms is essential when you're working with differences of logarithms. This rule states that subtracting one logarithm from another is the same as taking the logarithm of their quotient. In formulaic terms: \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \).

Here's how you apply the quotient rule in practice:

• Suppose you have the expression \(\ln(x^2 - 1) - \ln((x^2 + 3)^2)\).
• By the quotient rule, you combine them into a single logarithm: \(\ln \left( \frac{x^2 - 1}{(x^2 + 3)^2} \right)\).
• This simplification makes the expression more manageable and is especially useful in solving equations and comparing logarithmic quantities.

Mastering the quotient rule empowers you to handle subtractions within logarithmic functions effectively, streamlining your mathematical work.