Problem 41 - Calculus with Concepts in Calculus

Question

Let \(f(x)=\ln (4 x)-\ln x^{3}+\ln x^{2} .\) Plot \(f\) on a graphics calculator, and use properties of logarithms to explain the appearance of the graph.

Step-by-Step Solution

Answer

The graph is a horizontal line at \(f(x) = \ln(4)\), showing the function is constant.

Step 1: Apply Properties of Logarithms

Begin by simplifying the expression for \(f(x)\). Use the properties of logarithms: \(\log_a (b) - \log_a (c) = \log_a \left( \frac{b}{c} \right)\) and \(\log_a (b^c) = c \cdot \log_a (b)\). The given expression is \(f(x)=\ln (4x)-\ln x^{3}+ ln x^{2}\). Start by combining the first two terms: \(\ln (4x) - \ln x^3 = \ln \left( \frac{4x}{x^3} \right) = \ln \left( \frac{4}{x^2} \right)\).

Step 2: Simplify Further

Now combine \(\ln \left( \frac{4}{x^2} \right)\) with \(\ln x^2\). Using the properties of logarithms, we get: \(\ln \left( \frac{4}{x^2} \right) + \ln x^2 = \ln \left( \frac{4}{x^2} \cdot x^2 \right) = \ln(4)\). Hence, the function simplifies to a constant \(f(x) = \ln(4)\).

Step 3: Interpret the Graph

Since \(f(x) = \ln(4)\) is a constant function, the graph of \(f\) on a calculator is a horizontal line at the value \(\ln(4)\). This indicates that for every input \(x\), the output value does not change and is \(\ln(4)\), making it independent of \(x\).

Step 4: Plot and Verify

Plot \(f(x) = \ln(4)\) on a graphing calculator. You should see a horizontal line. Confirm that it matches the appearance expected from the algebraic manipulation, which shows that the function is a constant.

Key Concepts

Properties of Logarithms

Logarithms have specific rules that help simplify expressions, making them easier to interpret and solve. Key properties include:

• **Product Property**: \[\log_a(b) + \log_a(c) = \log_a(bc)\]This property shows that the sum of two logarithms is the logarithm of the product of their arguments.


• **Quotient Property**: \[\log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right)\]This means the difference of two logarithms can be rewritten as the logarithm of a division.


• **Power Property**: \[\log_a(b^c) = c \cdot \log_a(b)\]This states that the logarithm of a power can be expressed as the exponent times the logarithm of the base.

In the original exercise, these properties are used to simplify the function \(\ln(4x) - \ln x^3 + \ln x^2\) to a constant \(\ln(4)\), demonstrating how each term can be combined effectively.

Graphing Calculators

Graphing calculators are powerful tools that allow us to visualize mathematical functions easily. By providing a visual representation, these calculators help deepen your understanding of functions and their behaviors.

To plot a function like \(f(x) = \ln(4)\), enter the expression into the calculator and observe the graph. Since \(\ln(4)\) is a constant, its graph is a straight, horizontal line, indicating that the output value remains the same irrespective of the input \(x\).

Using a graphing calculator can:

• Confirm algebraic solutions visually
• Allow quick comparisons between different functions
• Help identify important characteristics of functions

Plotting functions helps verify algebraic work and provides insight into the behavior and characteristics of functions, whether they change or remain constant.

Constant Functions

A constant function is one where the output value is the same, no matter what input value is used. This results in a horizontal line on the graph, which is evident when plotting \(f(x) = \ln(4)\).

Characteristics of constant functions include:

• No variation in output: Regardless of the input, the output remains fixed. This is why the graph is a horizontal line at \(y = \ln(4)\).
• Slope of zero: Since the function does not rise or drop, the slope, which measures steepness, is zero.
• Simplicity in calculation: Since the value does not depend on \(x\), computations become straightforward.

Understanding constant functions is essential because they represent fundamental elements in mathematics and set a base for more complex topics. They serve as benchmark functions against which variables' behavior in other functions can be compared.