Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{1 / 4} x^{2}$$.

Step-by-Step Solution

Answer

The function \(f(x)=\log _{1 / 4} x^{2}\) transforms into \(f(x) = -2\ln(x) / \ln(4)\) after applying the change-of-base formula. The graph of \(f(x) = -2\ln(x) / \ln(4)\) is a reflection of \(y = ln(x)\) over the x-axis, stretched vertically by a factor of \(2\), and scaled vertically by \(-1 / \ln(4)\).

Step 1: Application of Change of Base

The change-of-base formula is \(\log _{a} x=(\ln x) /(\ln a)\). Apply it to \(f(x)=\log _{1 / 4} x^{2}\) to change the base of the logarithm. This transforms the function to \(f(x) = \ln(x^{2}) / \ln(1/4)\).

Step 2: Simplification

Now, we simplify the expression. Observe that the natural log of \(x^{2}\) is \(2\ln(x)\). Also, the natural log of \(1/4\) is \(-\ln(4)\). Therefore, \(f(x) = 2\ln(x) / -\ln(4) = -2\ln(x) / \ln(4)\)

Step 3: Graphing the Function

Use a graphing utility to plot \(f(x) = -2\ln(x) / \ln(4)\). This is a transformed version of the natural logarithmic function and should appear as a reflection of \(y = ln(x)\) over the x-axis, stretched vertically by a factor of 2, with a cortical scaling of \(-1 / \ln(4)\).

Key Concepts

Graphing Logarithmic Functions

Graphing logarithmic functions is a key skill in understanding the behavior of logarithms. To graph a logarithmic function, such as the given exercise function, we start with the basic logarithmic curve and apply transformations such as reflections, stretches, or compressions. The function in our example, \(f(x) = \log _{1 / 4} x^{2}\), can be graphed using a graphing utility after applying the change-of-base formula.

Through the formula, we converted the base to the natural logarithm, which allows the use of calculators and computer software as they typically have built-in functions to calculate natural logarithms, represented as \(\ln\). Knowing how to transform and graph these functions visually displays the behavior of logarithms and is crucial for deeper comprehension of the underlying mathematics.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. The natural logarithm is significant due to its intrinsic appearance in growth and decay models, compounded interest calculations, and in various branches of mathematics and physics.

In the context of the exercise, we use the natural logarithm to convert the base of the original logarithmic function from 1/4 to \(e\), facilitating the use of graphing utilities. This underscores the versatility of the natural logarithm as a tool for simplifying and calculating complex logarithmic expressions.

Logarithmic Properties

The properties of logarithms play a critical role when transforming and simplifying logarithmic expressions. Key properties include the product, quotient, and power rules.

• The product rule states that \(\log_b(mn) = \log_b(m) + \log_b(n)\)
• The quotient rule says that \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\)
• The power rule dictates that \(\log_b(m^n) = n\log_b(m)\).

In the exercise, we used the power rule by converting \(\ln(x^2)\) to \(2\ln(x)\), highlighting how the power rule simplifies the original expression and aids in the transformation process.

Logarithmic Transformation

Logarithmic transformations include various operations that reshape the graph of a function. For instance, multiplying by a negative factor reflects the function across the x-axis, and changing the coefficient of \(\ln(x)\) stretches or compresses the graph vertically.

In the solution provided, the expression \(f(x) = -2\ln(x) / \ln(4)\) suggests both a reflection and a vertical stretch. Specifically, multiplying by -2 reflects the graph of \(y = \ln(x)\) across the x-axis, while dividing by \(\ln(4)\) affects the horizontal scaling. These transformations provide a clear visual representation of how the function behaves and can help students predict the shape of the graph before using a graphing utility.