Problem 43 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\ln \left(x^{2}-1\right)-\ln (x-1)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Formatting Rule

Since we have two logarithms subtracted, we will use the logarithm property: \(\ln a- \ln b= \ln \frac{a}{b}\). Here, \(a = x^{2} - 1\) and \(b = x - 1\)

Step 2: Apply the Quotient Rule

We apply the quotient rule, rewriting the subtraction of the two natural logarithms as a single natural logarithm of their quotient: \(\ln \left(x^{2}-1\right)- \ln (x-1) = \ln\left(\frac{x^{2}-1}{x-1}\right)\)

Step 3: Simplify the Fraction

By simplifying the fraction, we come across a common factor in the numerator and the denominator: \(\frac{x^{2}-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1\).

Step 4: Substitute Back into the Logarithm

We substitute the simplified version of the quotient back into the logarithm, which results in: \(\ln(x+1)\).

Key Concepts

Natural Logarithm

The natural logarithm, often represented as "ln," is a special logarithmic function with a base of "e." The constant "e" is an irrational number approximately equal to 2.71828. Natural logarithms are widely used in solving mathematical problems due to their connection with the rate of growth and decay processes.
In the original exercise, we deal with the natural logarithms \( \ln(x^2-1) \) and \( \ln(x-1) \). When subtracting these logarithms, we make use of specific properties that simplify calculations, allowing for the expression to be compacted into a single natural logarithm.
Familiarity with the natural logarithmic functions can significantly enhance your understanding of mathematical growth patterns and help in simplifying expressions efficiently.

Quotient Rule

The quotient rule is a vital property in logarithmic expressions. It states that the difference between two logarithms, like \( \ln a - \ln b \), can be combined into a single logarithm: \( \ln \left( \frac{a}{b} \right) \). This is beneficial for condensing expressions and solving logarithmic equations with ease.

• In our example, \( a \) is \( x^2-1 \), and \( b \) is \( x-1 \).
• Applying the quotient rule, we rewrite the expression: \( \ln (x^2-1) - \ln (x-1) = \ln \left( \frac{x^2-1}{x-1} \right) \).

Using the quotient rule not only simplifies expressions but can also make solving equations more straightforward.

Simplification of Expressions

Simplifying expressions is an essential step in solving mathematical problems, especially when working with logarithmic functions. It involves reducing expressions to their simplest form, often by eliminating common factors or applying algebraic identities.
In the solution, simplifying \( \frac{x^2-1}{x-1} \) leads to a significant reduction in the complexity of the expression.

• This fraction simplifies to \( x+1 \) by recognizing and canceling the common factor \((x-1)\) in the numerator and denominator.
• The final simplification substitutes back into the logarithmic function to give \( \ln(x+1) \).

Understanding how to simplify expressions not only aids in solving problems faster but also enhances the clarity and elegance of mathematical solutions.