Problem 42 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, write the expression as the logarithm of a single quantity. $$ \ln (2 x+1)+\ln (2 x-1) $$

Step-by-Step Solution

Answer

The expression, written as the logarithm of a single quantity, is \(\ln [4x^2 - 1]\)

Step 1: Identification of the rule

The given expression contains two logarithms with the same base being summed. Therefore, the product rule of logarithms, which states that the sum of two logarithms with the same base is equal to the logarithm of the product of the contents of the two logarithms, can be applied.

Step 2: Apply the product rule

Apply the product rule \(ln (x) + ln (y) = ln (xy)\) to the given expression \(\ln (2 x+1) + \ln (2 x-1)\). This results in \(\ln [(2x+1)*(2x-1)]\).

Step 3: Simplify the expression

Simplify the expression inside the logarithm [(2x+1)*(2x-1)]. This results in \(\ln [4x^2 - 1]\).

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a fundamental concept used to simplify logarithmic expressions that involve the addition of logarithms with the same base. According to this rule, if you have two logarithms added together, like \( \log_a{(x)} + \log_a{(y)} \) you can combine them into a single logarithm by multiplying their arguments. Mathematically, this is represented as \( \log_a{(x)} + \log_a{(y)} = \log_a{(xy)} \).

In our textbook problem, we see this rule in action. We have two natural logarithms (logarithms with base \( e \) which is approximately 2.71828) being summed: \( \ln (2x+1) + \ln (2x-1) \). By applying the product rule, we end up with a single logarithmic expression: \( \ln [(2x+1)(2x-1)] \), which is equivalent to \( \ln [4x^2 - 1] \). Understanding and applying this rule correctly allows students to simplify complex logarithmic expressions to a form that is often easier to handle in further calculations or applications.

Logarithmic Expressions

Logarithmic expressions can be manipulative in different ways to help us solve equations and simplify calculations involving exponential relationships. Essentially, a logarithmic expression represents the exponent to which the base must be raised to produce a certain number. It comes in the form of \( \log_b{(a)} = c \), which means \( b^c = a \).

When dealing with logarithmic expressions, it's crucial to understand the properties and rules that govern them, like the product rule we already discussed. Additionally, there are the quotient and power rules that also aid in simplifying logarithms or rewriting them in different forms for ease of calculation or interpretation.

Real-World Application

In real life, logarithmic expressions are used in measuring the intensity of earthquakes (Richter scale) or the acidity of a substance (pH scale), illustrating their importance beyond just mathematical exercises.

Simplifying Logarithms

Simplifying logarithms involves rewriting logarithmic expressions in their simplest form using the rules and properties of logarithms. This can be an advantageous skill when trying to solve logarithmic equations or when you are interested in calculating the value of a logarithmic expression.

Apart from the product rule that we've discussed, other rules such as the quotient rule \( \log_a{(x/y)} = \log_a{(x)} - \log_a{(y)} \) and the power rule \( \log_a{(x^b)} = b \cdot \log_a{(x)} \) are also pivotal in the simplification process.

In our textbook problem after the application of the product rule, the expression is simplified further by expanding the multiplication \( (2x+1)(2x-1) \) to get \( 4x^2 - 1 \). The ability to recognize and apply these rules allows students to confidently navigate through problems involving logarithmic expressions and find solutions more efficiently.