Question

In Problems 79-84, graph each function using a graphing utility and the Change-of-Base Formula. \(y=\log _{4} x\)

Short answer

Use \ y = \frac{\log_{10} x}{0.602} \ to graph \( y = \log_{4} x \) using a graphing utility.

Step-by-step solution

- Understand the function

The function given is \( y = \log_{4} x \). This represents a logarithmic function with base 4.

- Use the Change-of-Base Formula

The Change-of-Base Formula for logarithms is \[ \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \]. Using base 10 (common logarithms), we can rewrite the function as \[ y = \frac{\log_{10} x}{\log_{10} 4} \].

- Simplify the Expression

Evaluate \[ \log_{10} 4 \]. Using a calculator, \log_{10} 4 \approx 0.602. Therefore, the equation simplifies to \[ y = \frac{\log_{10} x}{0.602} \].

- Graph the Function Using a Graphing Utility

Input the simplified function \[ y = \frac{\log_{10} x}{0.602} \] into a graphing utility. Generate the graph to visualize the behavior of the function.

- Analyze the Graph

The graph of the function \( y = \log_{4} x \) will show a logarithmic curve starting at (1,0) and passing through other key points such as (4,1). As x approaches infinity, y increases slowly. As x approaches 0 from the right, y decreases without bound.

Key concepts

Logarithmic Function

A logarithmic function is a mathematical function that involves the logarithm of a variable. The general form of a logarithmic function is given by \(y = \log_{b}(x)\), where \(b\) is the base of the logarithm and \(x\) is the input variable. In the given exercise, the function is \(y = \log_{4}(x)\), which means we're working with a base 4 logarithm.

Graphing Utility

A graphing utility is a tool, such as a graphing calculator or computer software, used to visualize mathematical functions. To graph the function \(y = \log_{4}(x)\) using a graphing utility, follow these steps:
- First, use the Change-of-Base Formula to convert the logarithm to base 10: \[y = \frac{\log_{10}(x)}{\log_{10}(4)}\]\.
- Simplify the expression by evaluating \(\log_{10}(4)\) using a calculator. This value is approximately 0.602.
- Input the simplified function, \[y = \frac{\log_{10}(x)}{0.602}\], into the graphing utility.
- Generate the graph to observe the behavior of the function.

Logarithms

Logarithms are the inverse operations of exponentiation. If \(b^{y} = x\), then \(y = \log_{b}(x)\). The logarithm \(\log_{b}(x)\) asks the question: 'To what power must the base \(b\) be raised, to result in \(x\)?' In the case of base 4, \(y = \log_{4}(x)\) is telling us the power to which 4 must be raised to obtain the value \(x\). Understanding logarithms makes it easier to solve exponential equations and many practical problems in science and engineering.