Problem 105 - Precalculus with Limits

Question

Using the Change-of-Base Formula In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 4} x$$

Step-by-Step Solution

Answer

The function \(f(x) = \log_{1/4}x\) can be rewritten using the change-of-base formula as \(f(x) = \frac{\log_{10} x}{\log_{10} (1/4)}\). A graphing utility can then be used to visualize this function.

Step 1: Understanding the Change-of-Base Formula

The Change-of-Base formula is used to rewrite logarithms in terms of any base of logarithm you want. The formula is \(\log_b a = \frac{\log_c a}{\log_c b}\). Here, \(c\) could be any positive number, but typically you use 10 or \(e\) since calculators and computers have built-in functions for \(\log_{10}\) and \(\log_e\) or ln.

Step 2: Apply the Change-of-base formula

We can use this formula to rewrite the logarithm \(f(x) = \log_{1/4}x\) as a ratio of logarithms with a new base, let's say, 10. By substitifying \(a=x\), \(b=1/4\), and \(c=10\) into formula, we get \(f(x) = \frac{\log_{10} x}{\log_{10} (1/4)}\).

Step 3: Graph the Function

Use a graphing utility to plot the function \(f(x) = \frac{\log_{10} x}{\log_{10} (1/4)}\). The graph will help visualize how the function behaves over its domain.

Key Concepts

The Concept of Logarithms

Logarithms help us understand the inverse relationship between exponentiation and multiplication. They answer the question: "To what power must a certain number, called the base, be raised to produce another number?" For example, in the equation \( \log_b a = x \), \(x\) is the logarithm of \(a\) with base \(b\). This equation says \(b\) raised to the power of \(x\) equals \(a\). So, if \( b^x = a\), then \( \log_b a = x \).

There are a few important bases to remember:

• Base 10: Known as the common logarithm (\(\log\) or \(\log_{10}\))
• Base \(e\) (around 2.718): The natural logarithm (\(\ln\))
• Any other base, such as \( \log_2 \) or \( \log_{1/4} \)

The change-of-base formula is a convenient tool for converting logarithms from one base to another, particularly to base 10 or \(e\), which are commonly used in calculators. It allows us to understand and calculate logarithms more easily using available computational tools.

Utilizing a Graphing Utility

A graphing utility is an invaluable tool in mathematics, allowing us to visualize and explore functions. These utilities can be devices like a graphing calculator or software applications, often capable of plotting complex mathematical functions on a graph.

To use a graphing utility for a logarithmic function like \( f(x) = \frac{\log_{10} x}{\log_{10} (1/4)} \), follow these steps:

• Enter the transformed equation into the graphing utility. Most accept input in the form of \( \log(x) \) or \( \ln(x) \).
• Set up the viewing window. Make sure it covers the domain of the function you're examining.
• Observe the graph. Notice how it changes when you adjust the window or input parameters.

Graphing utilities help visualize the behavior of functions, offering insight into where they increase or decrease, where they converge to zero, and other characteristics like intercepts and asymptotes. They are essential for gaining a deeper understanding of the mathematical concepts at play.

Understanding Logarithmic Functions

Logarithmic functions, such as \( f(x) = \log_b x \), are a special type of mathematical function that are the inverses of exponential functions. Their key characteristics include:

• Domain: Logarithms are not defined for non-positive numbers, so the domain of \( f(x) = \log_b x \) is \( x > 0 \).
• Range: The range of a logarithmic function is all real numbers.
• Intercept: The graph of \( \log_b x \) will always pass through the point (1,0), since \( \log_b 1 = 0 \).
• Growth: Logarithmic functions grow slower than linear functions and much slower than exponential functions.

When using the change-of-base formula to analyze these functions, it's important to recognize that the resulting quotient of logarithms reflects the same growth properties, just scaled depending on the bases used. Using a tool like a graphing utility can make these relationships clearer and reinforce understanding of how changes in base affect function behavior.